Elastic Shear Modulus Research Articles

Mechanical characterization and constitutive modeling of additively-manufactured polymeric materials and lattice structures

Additively manufactured polymeric lattice structures are being extensively studied, primarily because their mechanical properties can be tailored by controlling the unit cell geometry, giving them higher designability than stochastic materials. However, the inherent layer-wise additive manufacturing process affects the base material properties related to the printing direction, which in turn affects the macroscopic responses of the entire lattice materials. A robust understanding and modeling of lattice structures' elastic and plastic yield behavior in a homogenized approach are essential to enhance their design and analysis efficiency in engineering applications. In pursuit of this goal, a unified printing angle-dependent constitutive model of base materials is proposed in line with the tensile experimental data. The elastic material properties (elastic modulus, shear modulus, and Poisson's ratio), obtained through numerical simulations of one unit-cell with periodic boundary conditions, exhibit anisotropic properties, with the degree of anisotropy determined by the angle of the constituent members and base materials. Furthermore, both experimental and numerical results of lattices demonstrate anisotropic mechanical response under horizontal and vertical compression. Virtual multiaxial experiments are conducted through multi-cell numerical simulations, enabling the determination of initial yielding points of two different lattice structures (Kelvin and Simple cubic and body-centered cubic hybrid structures) under various loading conditions using a dissipation energy-based criterion. Overall, the multiaxial yield surface of the investigated lattices under various stress states, except for the isotropic principal stress plane, can be properly depicted by the Extended-Hill anisotropic yield criterion.

Destabilization Mechanism and Stability Study of Collapsible Loess Canal Slopes in Cold and Arid Regions

The combination of seasonal shutdowns, water conveyance, cold, and drought can easily lead to the deterioration of the anti-seepage system and loess foundation of the canal, which contributes to the destruction of the slope. To reveal the failure mechanism of the collapsible loess canal slope, this paper is based on the results of laboratory tests and adopts numerical simulations to analyze the stability of the canal slope under different conditions. The results show that the shear strength indexes and elastic modulus E of loess decrease following an exponential pattern with the increase in wetting-drying and freezing-thawing (WD-FT) cycles. The height of the seepage overflow point yields little effect on the water level behind the impermeable membrane, whereas the height of the water level has a significant effect. In the operation period, the slope under any working conditions is in a relatively stable state. However, the slope with a water level of 4.5 m behind the impermeable membrane tends to be unstable after three WD-FT cycles during the shutdown period. By replacing the surface-degraded loess with sand gravel and picking a depth of 0.9–1.2 m, the slope will maintain a long-term stable state.

Coupled effects of temperature and suction on the anisotropic elastic shear moduli of unsaturated soil

Elastic shear moduli of soil at various temperatures and suctions are important for analysing the serviceability limit state of energy piles and many other structures. Up to now, however, the coupled effects of temperature and suction on elastic shear modulus and the stiffness anisotropy, have not been well understood. This experimental study investigated the anisotropic elastic shear modulus of a compacted lateritic clay. A temperature and suction-controlled triaxial apparatus equipped with bender element probes and local strain measurements was used. Soil suctions from 0 to 300 kPa, and a temperature range of 5–40 °C were applied. The results at saturated and unsaturated conditions consistently reveal that the shear modulus is smaller after heating at a given stress and suction. Several mechanisms may contribute to this thermal-induced reduction in shear modulus, such as the heating-induced reduction of interparticle force and air–water surface tension. Moreover, the reduction in shear modulus upon heating depends on the shear plane and the degree of anisotropy changes.

Investigation of punching shear performance in concrete slabs reinforced with GFRP and synthetic fibers: An experimental study

The punching shear resistance mechanism of reinforced concrete slabs with glass fiber-reinforced polymers (GFRP-RC) is weaker than that of steel-RC slabs. The reduced shear strength and modulus of elasticity of GFRP reinforcement which led to greater tensile strains of the reinforcement and shallower compressive zone reduce the contribution of concrete in compression, aggregate interlocking, and dowel action to the punching shear mechanism. This condition results in reduced capacity and a higher likelihood of catastrophic failure compared to their steel-RC counterparts. To address this weakness of GFRP-RC, this paper aims to examine the punching shear performance of GFRP-RC slabs and explore the potential improvements through the incorporation of synthetic fibers. A total of 11 two-way slabs having a dimension of 1000 × 1000 mm with a thickness of 80 mm were tested under punching shear load and the effect of several types of fiber (polypropylene (PP) and (polyvinyl alcohol (PVA)), fiber content (0, 0.5%), reinforcement ratio, and reinforcement type (steel and GFRP) was investigated. The slabs’ behavior was assessed by considering crack width, deflection performance, the mode of failure, and the overall load-bearing capacity. The findings indicated that the influence of fibers was more noticeable in GFRP-RC slabs compared with steel-RC counterparts. Moreover, the fibers were found to effectively reduce deflections and crack widths. The results particularly showed the superior performance of PVA fibers in improving the capacity of the two-way slabs, whereby the addition of 0.5% PVA fibers led to the punching shear capacity increase of 44–51%. The findings of this study offer a better understanding of the punching shear behavior of slabs reinforced with GFRP reinforcement and synthetic fibers.

A comprehensive study on predicting the elastic modulus and Poisson ratio of hardened cement pastes via micro-scale cement hydration simulations

A comprehensive study was conducted to predict the elastic modulus and Poisson ratio of hardened cement pastes (HCPs), including numerical work, comparison of two modelling approaches, and experimental validation. Modelling approach 1 (M1) applied a continuum micromechanics model (CMM) whereas modelling approach 2 (M2) used finite element analysis (FEA). This study presents a reliable method for predicting the mechanical properties of HCPs by firstly employing the microstructures of HCPs, obtained by CEMHYD3D analysis with random distribution of cement particles as input parameters for both M1 and M2 approaches. The elastic modulus and Poisson ratio predicted by M1 and M2 approaches were verified through the experimental data obtained from compressive tests combined with digital image correlation analysis. The elastic moduli of HCPs were significantly affected by the input C-S-H modulus, with 90% of reported C-S-H elastic moduli falling within the range of 20.25 GPa and 30.4 GPa. The predicted elastic moduli of HCPs were well-matched with experimental data when assigning the elastic modulus, bulk modulus, and shear modulus, of C-S-H, as 20.25, 12.98, and 8.17 GPa, respectively. A sensitivity analysis demonstrated a linear increase in the elastic modulus of HCPs with increase of C-S-H modulus, with correlation coefficients R² of 0.9985 and 0.9996 for HCPs with water-to-cement ratios (w/c) of 0.3 and 0.5, respectively. Additionally, the predicted results showed greater consistency at later age (after 7 d) compared to earlier stages. The integral absolute error values of the elastic moduli obtained via M1 and M2 for HCPs with w/c of 0.3 were 6.32% and 6.23%, respectively while those for HCPs with w/c of 0.5 were 4.8% and 5.67%, respectively. M1 approach is more recommended since it is straightforward with time and cost efficiency compared to M2 approach in addition to the suitable prediction accuracy.

The effect of combined use of plybamboo sheathings and self-tapping screws in enhancing wood-frame shear wall performance

ABSTRACT Plybamboo panels are renewable and eco-friendly structural panels with high strength, good dimensional stability, and affordability, making them highly suitable for wood-frame construction. This study presents an experimental investigation of the wood-frame shear wall constructed with plybamboo-sheathing panels with self-tapping screws (STSs) subjected to monotonic and cyclic loadings. Performance parameters including shear strength, elastic shear modulus, ductility ratio, stiffness degradation, and energy dissipation were evaluated. Results indicate that the combined use of plybamboo sheathings and STSs can substantially reduce failures due to the tearing of sheathing, fastener head pull-through, and fastener withdrawal. The plybamboo-sheathed wood-frame shear walls with STSs had over 1.8 times shear strength and 1.4 times ultimate displacement than those of comparable wood-frame shear walls with nails, respectively, maintaining over 80% of their maximum lateral load capacity at a drift level over 4%. Moreover, the influence of aspect ratio and screw spacing on the elastic shear modulus, shear strength, and stiffness degradation of the plybamboo-sheathed wood-frame shear walls with STSs are consistent with those for common wood-frame shear walls.

Intramuscular fat infiltration influences mechanical properties during muscle contraction in older women.

Although evidence suggests that intramuscular fat infiltration may influence muscle strength, the precise mechanisms remain unclear. This study aimed to determine whether intramuscular fat infiltration affects muscle mechanical properties during contraction and whether these mechanical properties mediate the relationship between intramuscular fat infiltration and muscle strength. Seventy-nine healthy older women aged 75.1±6.8 years were included in this study. The echo intensity (EI) of the vastus lateralis (VL) was measured as an intramuscular fat infiltration index using B-mode ultrasonography. Maximum voluntary isometric contraction strength (MVIC) was assessed using a dynamometer. The VL shear elastic modulus (G), a mechanical property index, was measured using ultrasound shear wave elastography under various muscle contraction conditions, at rest and at 15%, 30%, and 45% MVIC (G0, G15, G30, and G45). To evaluate the degree of increase in the shear elastic modulus with increasing muscle contraction intensity, the slope of the regression line (Gslope) between muscle contraction and shear elastic modulus was calculated for each participant. The results showed that EI was significantly associated with G30 and G45 but not with G0 or G15. The EI can significantly explain the inter-individual differences in Gslope. Mediation analysis revealed that the effect of EI on MVIC through Gslope was significant (indirect effect = -0.31, 95% confidence interval (-0.57, -0.12)). These findings suggest that a greater EI is associated with a lower G during muscle contraction. Furthermore, our results show that the relationship between EI and MVIC is mediated by Gslope.

Analytical calculation model for elastic modulus and strength of foamed concrete

Foamed concrete has gained increasing attention for its excellent engineering performance. Endeavors have been made to experimentally and numerically elucidate the mechanism of foamed concrete, while few theoretical methods are used to analytically calculate mechanical properties. This paper proposes a theoretical framework to describe the elastic behavior of foamed concrete by combining Eshelby’s equivalent inclusion theory with the Mori–Tanaka method. In the analytical model, foamed concrete is regarded as a two-phase composite comprising a homogeneous cement matrix and randomly oriented ellipsoidal air void. During derivation, the elastic modulus, strength, bulk modulus, shear modulus, and Poisson ratio are measured. A good agreement is established between the proposed method and experimental data.

Effective Stretching Positions of the Piriformis Muscle Evaluated Using Shear Wave Elastography.

Piriformis syndrome is often associated with muscle spasms and shortening of the piriformis muscle (PM). Physical therapy, including static stretching of the PM, is one of the treatments for this syndrome. However, the effective stretching position of the PM is unclear in vivo. This study aimed to determine the effective stretching positions of the PM using ultrasonic shear wave elastography. Observational study. Twenty-one healthy young men (22.7 [2.4]y) participated in this study. The shear elastic modulus of the PM was measured at 12 stretching positions using shear wave elastography. Three of the 12 positions were tested with maximum internal rotation at 0°, 20°, or 40° hip adduction in 90° hip flexion. Nine of the 12 positions were tested with maximum external rotation at positions combined with 3 hip-flexion angles (70°, 90°, and 110°) and 3 hip-adduction angles (0°, 20°, and 40°). The shear elastic modulus of the PM was significantly higher in the order of 40°, 20°, and 0° of adduction and higher in external rotation than in internal rotation. The shear elastic modulus of the PM was significantly greater in combined 110° hip flexion and 40° adduction with maximum external rotation than in all other positions. This study revealed that the position in which the PM was most stretched was maximum external rotation with 110° hip flexion and 40° hip adduction.

Effects of high-frequency hyperthermia on the elastic modulus of the lumbar muscle in female athletes with low back pain: A randomized crossover trial.

To investigate the effects of capacitive and resistive monopolar radiofrequency (CRMF) on the shear elastic modulus of the multifidus and erector spinae muscles in female athletes with low back pain (LBP) and a history of LBP. Randomized crossover trial. Academic institution. Twenty female university athletes with LBP or a history of LBP were included. All participants received CRMF, hotpack, and sham (CRMF without power) in a random order on the right side of the lumbar region. More than 2 days were allocated between the experiments to eliminate any residual effects. The shear elastic moduli of the right multifidus and erector spinae were evaluated in the prone (rest) position while sitting with 35° trunk flexion (stretched) using shear wave ultrasound imaging equipment. The moduli were measured before, immediately after, and 30 minutes after the intervention. Repeated-measures 2-way analysis of variance and post hoc analysis showed that the moduli of the CRMF group were significantly lower than those of the sham group in the stretched position immediately after intervention (P = .045). This difference diminished 30 minutes after the intervention (P = .920). CRMF can be used to reduce the shear elastic modulus of the multifidus muscle in the short term. Further studies are warranted to determine how to provide longer effects. None.

The mechanical properties of Al3Sc single crystal investigated by nanoindentation using Hertzian elasticity theory method and Oliver-Pharr method

The crystal structure, orientation, reduced modulus, and hardness of Al3Sc single crystal were studied using X-ray diffraction (XRD), electron backscatter diffraction (EBSD) and nanoindentation techniques, combined with Hertzian elastic theory method (Hertz) and Oliver-Pharr method (O&P). Millimeter-sized Al3Sc single crystals were prepared using quasi-equilibrium solidification of hypereutectic Al-15(wt %) Sc alloy. The XRD results showed that the samples were composed of L12-Al3Sc and Al phases. The EBSD and nanoindentation results showed that for five Al3Sc single crystals with different orientations, the reduced moduli measured by Hertz method were 146.6 ± 0.4–148.3 ± 0.2 GPa; while measured by O&P method they were 147 ± 2–150 ± 3 GPa. The corresponding hardnesses were varied between 3.8 ± 0.1 and 3.9 ± 0.1 GPa. The Poisson's ratio and elastic constants were determined through iterative calculation based on the relationship between the reduced modulus and Young’s modulus at various orientations. The polycrystal elastic properties, bulk modulus, shear modulus and Young's modulus, were determined. The elastic constants c11, c12, and c44 were 179.2 ± 0.4, 44.1 ± 0.1, and 68.3 ± 0.3 GPa, respectively; the Poisson's ratio was 0.196 ± 0.001. The Young's modulus of five Al3Sc single crystals oriented in different directions was in the narrow range 161–164 GPa. The anisotropy factor of Al3Sc crystal was 1.011 ± 0.002, confirming that it is elastically isotropic. These results provide fundamental information for the research and development of the Al3Sc intermetallic compound and related alloys in future.

Influence of high pressure on mechanical and electronic properties of C3060 allotropes—A first-principles investigation

The influence of external uniaxial pressure, ranging from 0 GPa to 50 GPa, on (3,0)(6,0) carbon (C3060), is examined through first-principles calculation. The mechanical descriptors, including elastic constants, bulk modulus, shear modulus, and Young's modulus, are studied and reported. Moreover, these characteristics demonstrate the adaptable robustness of C3060 to mechanical stress under various pressure conditions. Also, the ductility and brittleness analysis is obtained based on Cauchy pressure, Poisson's ratio, and Pugh's criteria. The structural and dynamical stability of C3060 are ensured using cohesive formation energy, phonon band spectrum, and phonon density of states. Further, the thermodynamic properties of C3060 under high pressure are discussed with regard to Gibbs free energy and entropy. Also, the electronic properties of C3060 under pressure within the range of 0 GPa–50 GPa are explored based on the band structure and projected density of the state spectrum. In short, this work emphasizes how uniaxial pressure influences the mechanical characteristics and electronic attributes of C3060, providing information on the adaptability of the material's structural features with regard to mechanical attributes.

Estimating elastic properties of sediments by pseudo-wells generation utilizing simulated annealing optimization method

The hydrate concentration model considerably affects elastic properties, including bulk and shear modulus. Defining seismic properties of sediments, such as compressional and shear wave velocity and density, provides valuable information to identify rock facies and fluid types. This information commonly results from pre-stack seismic inversion, while post-stack seismic information provides acoustic impedance as a layer-based property. Traditionally, seismic inversion requires well logs to produce an initial guess of inversion routines and provide a low-frequency part of the amplitude spectrum. Eventually, seismic inversion methods could not be performed in the areas without well-log data, such as deep sea areas. In such cases, pseudo-well logs derived from pre-stack seismic data are a solution. Pseudo-well generation is a title used to estimate the elastic parameters of sediments in areas, such as deep marine environments, where drilled wells are absent or sparse. Metaheuristic optimization algorithms are suitable tools for minimizing the cost function as they best match real and synthetic seismic data. In this study, the SEAM earth model has been used as a reference to investigate the quality of pseudo-well generation utilizing a simulated annealing (SA) algorithm as an optimization method of property model change, which minimizes the cost function of seismic inversion. As a result, considering an initial model type of the SEAM model, simultaneous seismic inversion introduced the best compressional and shear wave velocities and density logs, which provide the best real and synthetic seismic data match when synthetic data is calculated from the simplified Zoeppritz equation.

Evolution of Elastic Shear Modulus During Shearing Under Different Stress Paths Considering Anisotropic Stress History

Evolution of Elastic Shear Modulus During Shearing Under Different Stress Paths Considering Anisotropic Stress History

Compaction and compressibility characteristics of snail shell ash and granulated blast furnace slag stabilized local bentonite for baseliner of landfill

This study comprehensively explores the compaction and compressibility characteristics of snail shell ash (SSA) and ground-granulated blast-furnace slag (GBFS) in stabilizing local bentonite for landfill baseliner applications. The untreated soil, with a liquid limit of 65%, plastic limit of 35%, and plasticity index of 30%, exhibited optimal compaction at a moisture content of 32% and a maximum dry density of 1423 kg/m3. SSA revealed a dominant presence of 91.551 wt% CaO, while GBFS contained substantial 53.023 wt% SiO2. Treated samples with 20% GBFS and 5% SSA exhibited the highest maximum dry density (1561 kg/m3) and optimal moisture content (13%), surpassing other mixtures. The 15% SSA-treated sample demonstrated superior strength enhancement, reaching an unconfined compressive strength of 272.61 kPa over 28 days, while the 10% GBFS-treated sample achieved 229.95 kPa. The combination of 15% SSA exhibited the highest shear strength (49 kPa) and elastic modulus (142 MPa), showcasing robust mechanical properties. Additionally, the 15% SSA sample displayed favourable hydraulic conductivity (5.57 × 10–8 cm/s), outperforming other mixtures. Notably, the permeability test, a critical aspect of the study, was meticulously conducted in triplicate, ensuring the reliability and reproducibility of the reported hydraulic conductivity values. Treated samples with SSA and GBFS showed reduced compressibility compared to the control soil, with the 15% SSA-treated sample exhibiting a more consistent response to applied pressures. Scanning Electron Microscopy analysis revealed substantial composition changes in the 15% SSA mixture, suggesting its potential as an effective base liner in landfill systems. In conclusion, the 15% SSA sample demonstrated superior mechanical properties and hydraulic conductivity, presenting a promising choice for landfill liner applications.

Validation of anterior ankle soft tissue dynamics and shear modulus for anterior ankle impingement syndrome after ankle fracture surgery

Anterior ankle impingement syndrome (AAIS) has been reported to account for a high percentage of complications following ankle fracture surgery. The soft tissue etiology of AAIS is thought to be thickening and inflammation of the anterior ankle soft tissues intervening anteriorly at the tibiotalar joint, causing pain and functional limitation during dorsiflexion. However, the effects of anterior ankle soft tissue dynamics and stiffness on AAIS have yet to be clarified. This study aimed to determine the relationship between AAIS and the anterior ankle soft tissue thickness change ratio and shear modulus using ultrasonography (US). The participants were 20 patients with ankle joint fractures (AO classification A, B) who had undergone open reduction and internal fixation and 20 healthy adults. The evaluation periods were 3 months and 6 months postoperatively. US was used to delineate the tibialis anterior tendon, extensor hallucis longus tendon, and the extensor digitorum longus tendon over the talus and tibia on a long-axis image. Anterior ankle soft tissue thickness was measured as the shortest distance from the most convex part of the talus to the tendon directly above it. The Anterior ankle soft tissue thickness change ratio was determined by dividing the value at 0° dorsiflexion by the value at 10° plantarflexion. The same images as for the anterior soft tissue thickness measurement were drawn for the shear modulus measurement, and the average shear modulus (kPa) was calculated using shear-wave elastography. There was no significant difference in the thickness change ratio between the postoperative and healthy groups. Compared with the healthy group, the shear modulus was significantly higher at 3 and 6 months in the postoperative group (p < 0.01). The shear elastic modulus at 6-month postoperative group was significantly lower than at 3-month postoperative group (p < 0.01). Anterior ankle joint soft tissue stiffness may increase after surgery for an ankle fracture.

Effect of Surface Chemistry on the Squeeze-Thin Film and Friction of Boundary Films.

An approach combining adsorption characterization and lubricity effectiveness of amine-based friction modifier molecules has been performed using chemically controlled surfaces, coated either with cobalt or carbon, while keeping the surface roughness constant and sub-nanometric. Through squeeze measurements and numerical modeling, we have identified the mechanical properties of both adsorbed amine films, as a function of the surface on which they were formed. On the one hand, we were able to evidence that the fluid structuring at the vicinity of the adsorbed boundary film differed as a function of the latter mechanical properties, directly resulting from its molecular organization. On the other hand, we showed that the coverage ratio of the monolayer associated with the shear elastic modulus of the boundary film governed the friction level. Changing the surface chemistry while keeping the roughness constant controls the final organization in the boundary layer, the correlated mechanical properties, and the level of friction dissipation.

Effect of repeated freeze-thaw cycles on mechanical properties of clay

To investigate the variation in the mechanical properties of clay under freeze-thaw cycles (FTCs), a series of experiments were conducted in the laboratory. Samples with different water contents and dry densities were subjected to FTCs ranging from 0 to 11 times. Then, cohesion, shear strength, internal friction angle and elastic modulus were obtained using triaxial test. The results show that with the increase in the number of FTCs, the shear strength, cohesion and elastic modulus decreased, while the internal friction angle increased slightly. However, the variation in the internal friction angle is not obvious, and the maximum increment is within 4°. The cohesion exhibited the most decrease after the first freeze-thaw action. Besides, under a same number of FTCs, four mechanical properties are significantly affected by water content and dry density. The shear strength, cohesion, elastic modulus and internal friction angle decrease with water content while increasing with dry density. Additionally, the elastic modulus is associated with confining pressure, which increases with confining pressure. This study provides evidence for the variation in mechanical properties of the soils subjected to FTCs and guides the design and construction of the cold regional engineering.

Strength, deformation, and environmental impact assessment of cement stabilized mine overburden soil

Mine haul roads are the unpaved roads that are constructed from the overburden materials obtained from the mining operations and are used for the movement of heavily loaded dumpers and trucks. The haul roads constructed from this overburden material shows continuous distress in the form of over ruts, potholes, and shear failures, creating major issues in the movement of dumpers. In the present research study, an experimental investigation was conducted based on mechanistic empirical design to evaluate the strength-deformation characteristics, durability, and environmental emissions of cement stabilized overburden soil of a local mine. The unconfined compression tests were conducted at cement dosages varying from 1 to 6% of dry soil mass and increase in the unconfined compressive strength were examined at different intervals of time until 28 days of the curing period. In the case of static triaxial tests, the enhancement in undrained shear strength and elastic modulus of stabilized specimens was determined in comparison to the untreated specimens, whereas in cyclic triaxial tests, the reduction in permanent deformations and increase in resilient modulus were evaluated under different confining pressures and cyclic deviatoric stresses. An empirical model has been proposed to predict the long-term rutting of overburden under repeated loading. The proposed model is based on the behaviour of various parameters, such as applied cyclic deviatoric stress, number of load repetitions, loading frequencies, and on the permanent deformation characteristic of mine overburden soil. In addition, durability and environmental sustainability aspects of the treatment has also been studied to determine the optimal dose of cement.

Measurement and Analysis of the Vibration Responses of Piano Soundboards with Different Structures.

The effect of structure on the vibration response was explored for four piano soundboards with different but commonly adopted structures. The vibration response was obtained using the free-vibration method, and the values of the dynamic modulus of elasticity and dynamic shear modulus obtained using the free-vibration frequency method (EF and GF) were compared with the dynamic modulus of elasticity obtained using the Euler beam method (EE) and dynamic shear modulus obtained using the free-plate torsional vibration method (GT), respectively. It was found that the soundboards with different structures had different vibration modes and that excitation at different locations highlighted different vibration modes. For all the soundboards analyzed, the EE and GT were higher than EF and GF by 2.2% and 24.3%, respectively. However, the trends of the results of these methods were the same. The four piano soundboards with different structures possessed varying dynamic moduli of elasticity and dynamic shear moduli. These rules are consistent with the grain directions of the soundboards and the anisotropy of the wood (the direction of the units of the soundboards). The results show that the vibration mode of the piano soundboard is complex. The dynamic elastic modulus of the soundboard can be calculated using the Euler beam method. The results provide a reference for studies on the vibration response, material selection, production technology, and testing of piano soundboards.