The shape of the cohesive arch in hoppers and silos — Some theoretical considerations

Models of the cohesive arch have been developed in order to predict arch shape and test the circular arc hypothesis first proposed by Enstad [‘On the theory of arching in mass flow hoppers’, Chem. Eng. Sci., 1975, 30, 10, 1273-1283].A 2-dimensional arch was modelled. Vertical and horizontal force balances described the system, with 2 unknown: arch stress, σ_arc, and vertical arch co-ordinate y. A rotationally symmetrical system was also modelled and included the azimuthal stress σ_az. σ_az was related to σ_arc by a Mohr-Coulomb yield type of equation. These equations were solved numerically by an Euler method.A least squares fit was used to predict the equivalent circular arc radius R and circular arc co-ordinate y_circ. The square of deviation from the circle Σ(y − y_circ)² was used as a statistical measure of the goodness of fit to the circular arc.The arch shape was generally an excellent approximation to the circular arc. However, the arch diverged from the circular as arch span increased.A dimensionless group was defined: the Stress-Radius Number, N_SR, which incorporates the ratio of equivalent circular arc radius to the arch stress at the apex. N_SR was constant for a given set of conditions and was ideally equal to 1 for a 2-dimensional arch and 2 for a rotationally symmetrical arch with no overpressure.Arch thickness models had little effect upon arch shape but had a great influence upon stress. This affected the critical outlet dimension for flow.Rotationally symmetrical arches were very sensitive to the relation between azimuthal and arch stresses. This affected both arch shape and stresses.A critical outlet dimension was calculated and showed great variation dependent upon assumptions made. Jenike's approach of an arch of constant thickness with no overpressure yielded a conservative value.     

Flow property analysis of irregular powders

Accurate yield locus analysis using computer techniques can now be carried out on simple and complex irregular powders, as well as on regular powders, by performing a least squares regression on the equation n log(τ_i/τ_L) = log [(T_s + σ_i/σ_L)/(T_s + 1)]This equation reduces any number of yield loci having a common shear index value, n, to a single straight line of slope n provided that all individual shear stress,τ_i, and normal stress, σ_i points are divided by their respective steadystate stress values τ_L and σ_L, and that the correct value for specific tension, T_s = T/σ_L is entered for each locus. Improved values for both n and T_s are found by successive reiteration of the above regression and by subsequent regeneration of individual yield locus values for cohesion, C, and tension, T, from a solution of the equation.

Simple irregular powders which show a constant n, but a variable angle of static internal friction, ψ_s, are easily handled by this technique. Complex irregular powders, in which both n and ψ_s vary, present greater difficulty, but can be analysed by taking a large number of points and operating on individual loci in pairs or one at a time. Examples are given of industrial powders showing both simple and complex irregularity, and particular reference is made to the effect of variable friction characteristics on overall flowability, ζ_i.     

Mechanical and thermal shock properties of size graded MgO–PSZ refractory

The effects of the mixture of coarse powder with fine PSZ powder on the thermal-mechanical properties of 10 Mg–PSZ samples were studied. The size graded specimens were injection-molded using 3.5 m% MgO–ZrO₂ powders. The physical properties of the ZrO₂ samples and five thermal shock parameters were measured and calculated. These properties included density (ρ), porosity (p), the ratio of m/(t+c+m) phase, fracture toughness (K_IC), strength (σ_f), Young's modulus (E), shear modulus (G), Poisson's ratio (ν), and the thermal expansion (α) between ambient temperature to 1100°C. The toughness and thermal shock resistance of the PSZ are controlled by the states of porous microstructure which can be represented by a parameter (nominal largest tolerable length of defects) a_t. The PSZ samples show two types of thermal shock behavior differentiated by comparing the value of a_t to the characteristic length L_f of the defects in the sintered PSZ. The states of the defects, i.e. porosity, are the microstructural evidence to explain the relationship between the thermal shock properties.     

Evaluation of stress distribution and failure mechanism in lanthanum–titanium–aluminum oxides thermal barrier coatings

LaTi₂Al₉O₁₉ (LTA) is one of the most promising materials for new thermal barrier coatings (TBCs) to fulfill the demand of advanced gas turbines owing to its high temperature stability and low thermal conductivity. In the present study, a finite element (FE) based numerical study has been carried out to investigate the stress distribution in LTA single layered coating system in comparison with traditional yttria stabilized zirconia (YSZ) TBC. Stresses in YSZ/LTA double ceramic layer TBC system are also determined and presented for comparative analysis. The thermal cycling effect is simulated by sequent increment in TGO thickness in a series of FE simulations. In-plane stresses (σ_xx), out-of-plane stresses (σ_yy) and shear stresses (σ_xy) are determined for all systems, and peak stress values are presented for quantitative comparison. Elastic strain energy stored in TGO of all systems is calculated from FE results for TBC structural integrity assessment. It has been found that maximum in-plane and shear stresses are lower in the double ceramic layer coating system than in the single layer ceramic coating system. However, peak axial tensile and compressive stresses in the double ceramic layer coating are very close or higher than those in the single layer topcoat. Calculation of elastic store energy shows that double ceramic layer TBC system may exhibit better stability as compared to single layer systems. Results are presented to explain the failure mechanism in LTA coatings.     

New criterion for craze initiation

Existing criteria for craze initiation are reviewed, and their limitations are discussed. The most obvious problem is that they are formulated simply in terms of principal stresses, making no provision for the known effects of small inclusions and surface imperfections. To solve this problem, a new criterion is proposed, which is based on linear elastic fracture mechanics. Craze initiation is treated as a frustrated fracture process rather than a yield mechanism. Calculations show that the strain energy release rate, G_I(nasc), required to generate a typical 20 nm thick nascent craze, is less than 1 J m⁻². This explains why flaws less than 1 μm in length are capable of nucleating crazes at stresses of 20-30 MPa. Subsequent craze propagation is dependent upon two flow rates, one relating to fibril drawing at the craze wall and the other to shear yielding at the craze tip. Under biaxial stress, the second principal stress σ₂ affects craze tip shear yielding but not fibril drawing. This model is used in conjunction with the von Mises yield criterion to derive a new expression for the crazing stress σ₁(craze), which provides a good fit to data on visible crazes obtained by Sternstein, Ongchin and Myers in biaxial tests on cast PMMA [Sternstein SS, Ongchin L, Silverman A. Appl Polym Symp 1968;7:175; Sternstein SS, Ongchin L. Polym Prepr Am Chem Soc Div Polym Chem 1969;10:1117; Sternstein SS, Myers FA. J Macromol Sci Phys 1973;B8:539].     

Slow and intermediate flow of a frictional bulk powder in the Couette geometry

Most current research in the field of dry, non-aerated powder flows is directed toward rapid granular flows of large particles. Slow, frictional, dense flows of powders in the so-called quasi-static regime were also studied extensively using Soil Mechanics principles. The present paper describes the rheological behavior of powders in the “intermediate” regime lying between the slow and rapid flow regimes. Flows in this regime have direct industrial relevance. Such flows occur when powders move relative to solid walls in hoppers, bins and around inserts or are mixed in high and low shear mixers using moving paddles. A simple geometry that of a Couette device is used as a benchmark of more complicated flows.The constitutive equations derived by Schaeffer [J. Differ. Equ. 66 (1987) 19] for slow, incompressible powder flows were used in a new approach proposed by Savage [J. Fluid Mech. 377 (1998) 1] to describe flows in the intermediate regime. The theory is based on the assumption that both stress and strain-rate fluctuations are present in the powder. Using Savage's approach, we derive an expression for the average stress that reduces to the quasi-static flow limit when fluctuations go to zero while, in the limit of large fluctuations, a “liquid-like”, “viscous” character is manifested by the bulk powder.An analytical solution of the averaged equations for the specific geometry of the Couette device is presented. We calculate both the velocity profile in the powder and the shear stress in the sheared layer and compare these results to experimental data. We show that normal stresses in the sheared layer depend linearly on depth (somewhat like in a fluid) and that the shear stress in the powder is shear rate dependent. We also find that the velocity of the powder in the vicinity of a rough, moving boundary, decays exponentially so that the flow is restricted to a small area adjacent to the wall. The width of this area is of the order of 10-13 particle diameters. In the limit of very small particles, this is tantamount to a shear band-type behavior near the wall.     

Mixing of pharmaceutical solids. I. Effect of particle size on mixing in cylindrical shear and V-shaped tumbling mixers

The mixing of a cohesive drug with a cohesive, non-cohesive and free-flowing excipient was studied using two types of mixers, cylindrical shear and V-shaped tumbling. Two mixing indices, one based on complete random mixing, s/σ_R, and the other based on standard specifications, s/σ_A, were used to evaluate the data. Both indices gave similar results for mixing cohesive drug with a free-flowing or non-cohesive excipient and were suitable for evaluating homogeneity. However, for mixing a cohesive drug with a cohesive excipient, s/σ_R was not a suitable index, while s/σ_A could be used. The ‘mixing margin’, a concept proposed by Hersey [3], is not a very useful tool in assessing mix-ability of powders having too large or too small particle size distributions. Because of electrostatic charging, preferential sticking of the drug to the walls of the mixer resulted in a lower mean value of the drug in the mixture. Although the mixing indices suggested that the desired mixedness was reached, it is proposed that the mean percent of the active ingredient should be checked in addition to the mixing indices for ensuring the uniformity and potency of the drug content in solid state mixtures.     

Elastic stresses in an adhesively-bonded functionally-graded tubular single-lap joint in tension

In this study, the elastic stress analysis of an adhesively-bonded tubular lap joint with functionally-graded Ni-Al₂O₃ adherends in tension was carried out using a 3D 8-node isoparametric multilayered finite element with 3 degrees-of-freedom at each node. Stress concentrations were observed along the edges of both outer and inner tubes in the overlap region. Thus, the outer tube region near the free edge of the inner tube and the inner tube region near the free edge of the outer tube experienced considerable stress concentrations. Normal σ_zz and shear σ_rz stresses were dominant among the stress components. In addition, both edges of the adhesive layer experience stress concentrations, and the von Mises σ _eqv stress decreases uniformly across the adhesive thickness at the free edge of the outer tube, whereas it increases at the free edge of the inner tube. However, different compositional gradients had only a small effect on the through-the-thickness normal and shear stress profiles of both outer and inner tubes, and the peak von Mises σ _eqv stresses occurred inside the tube walls. As the ceramic phase in the material composition of the outer and inner tubes was increased, peak von Mises σ _eqv stress appeared in the ceramic layer. However, its magnitude was increased 1.75-fold in both tubes. In addition, the peak adhesive stresses appeared at the edge of the outer tube–adhesive interface near the free edge of the inner tube and at the edge of the inner tube–adhesive interface near the free edge of the outer tube. Increasing the ceramic phase in the material composition caused 1.22–1.67-times higher von Mises stresses along the free edges of the adhesivetube interfaces. In addition, with increasing number of layers across the inner and outer tubes the profiles of the normal σ_zz , shear σ_r and von Mises σ _eqv stresses across the tube walls and adhesive layer become similar. Increasing the ceramic phase in the material composition of the tubes causes also evident increases in the normal σ_zz and von Mises stresses while it does not affect their through-the-thickness profiles. However, it affects only shear σ_r and von Mises stresses across the adhesive layer. Finally, the layer number and the compositional gradient do not affect considerably through-the-thickness normal and shear stress profiles but levels in a functionally graded plate subjected to structural loads.     

Stresses in a bulk solid in a cylindrical silo, including an analysis of ratholes and an interpretation of rathole stability criteria

A two-dimensional model for stresses in a cohesive bulk solid in a vertical, cylindrical vessel has been developed, assuming a rotated, circular arc orientation of principal stresses, after Enstad [1995. On the theory of arching in mass flow hoppers. Chemical Engineering Science 30 (10), 1273-1283] and Li [1994. Mechanics of arching in a moving bed standpipe with interstitial gas flow. Powder Technology 78, 179-187].The model assumes that principal stresses form a spherical dome surface within the vessel. At the wall, the surface makes a constant angle with the wall normal, necessitating a progression of arc centre up the vessel. An incremental element of upper and lower surfaces with vertical height δx at the wall has variable thickness with arc angle ε measured from the vessel centre. Positions within the vessel are located by height at which the arc cuts the vessel wall x, and arc angle ε. Rotational symmetry is assumed through azimuthal angle θ. This gives 3 principal stresses: σ_R; σ_ε;σ_θ.Static vertical and horizontal force balances yielded two partial differential equations, and a third was obtained by assuming a known ratio between effective stresses and . These equations were integrated numerically to give stress surfaces in x-ε space. The model simulated variations in the state of stress across the arc surface, and in some conditions, this lead to large variations in stress between the wall and vessel centre. The model accurately predicted the minimum outlet radius required for flow of a cohesive bulk material.The presence of a vertical, co-axial rathole was modelled with minor geometrical modifications. The rathole was assumed to be an unconfined plane of principal stress where the radial stress, σ_R, acted vertically downwards and its value increased linearly with bed depth, contrary to Jenike's [1964. Storage and flow of solids, Bull. 123, University of Utah, USA] assumption of a limiting stress in deep beds. Conventional stability criteria (σ_R<=f_c for stability), suggest that only shallow ratholes can exist. However, a modified yield criterion, allowing for the curvature of the rathole, enabled deep, stable ratholes to exist. Results show that the stability of deep ratholes depends upon the rathole radius and the vessel radius.     

Criteria to predict the embrittlement of polycarbonate

A direct correlation is found between the time evolution of the yield stress in unnotched tensile bars and that of the impact energy measured using notched tensile bars. In both cases a master curve can be constructed with an Arrhenius type of shift function, using the same activation energy. Combining these experimental findings with numerical simulations lead to a maximum hydrostatic stress as a criterion to predict the onset of failure, . Where aging kinetics is not dependent on the polymer’s molecular weight, embrittlement is, and for higher molecular weights a higher is found. Moreover, for high polymers we observe a more stable craze extension and crack propagation after cavitation, causing the computed impact energies to underestimate the experimental ones for all but the lowest molecular weight polymers. Because the maximum in hydrostatic stress under the notch, σ^h, increases with the polymer’s age, defined by the value of the state parameter S_a, and given the proportionality between S_a and the yield stress, σ_y, an alternative failure criterion is proposed which is a molecular weight dependent critical value of the evolving yield stress, σ_y,c. It predicts a product’s increased sensitivity to damage under the influence of progressive aging as enhanced by temperature.     

The measurement of powder flow properties with a mechanically stirred aerated bed

This paper re-examines a set of experimental data published by Bruni et al., 2007a, Bruni et al., 2007b [Bruni, G., Barletta, D., Poletto, M., Lettieri, P., 2007a. A rheological model for the flowability of aerated fine powders. Chem. Eng. Sci. 62, 397–407; Bruni, G., Lettieri, P., Newton, D., Barletta, D., 2007b. An investigation of the effect of the interparticle forces on the fluidization behaviour of fine powders linked with rheological studies. Chem. Eng. Sci. 62, 387–396] carried out on a mechanically stirred fluid-bed rheometer (msFBR), which was developed to study the rheology of aerated and fluidized powders. The use of aeration below fluidization allowed to carry out experiments with powders at very low consolidation levels. Two mathematical models, based on the Janssen approach to evaluate stresses in powder containers, were developed in order to relate the torque measurements in the Fluidized Bed Rheometer to the flow properties of the powders measured with standard powder flow testers. Results indicate that the models were able to satisfactorily predict the torque measured by the msFBR. The larger complexity of the Walker (1966) [Walker, D.M., 1966. An approximate theory for pressures and arching in hoppers, Chem. Eng. Sci. 21, 975–997] and Walters (1973) [Walters, J.K., 1973. A theoretical analysis of stresses in silos with vertical walls, Chem. Eng. Sci. 28, 13–21] stress analysis adopted in one of the two models did not introduce significant improvements in the evaluation of the stress distribution to justify its use. A procedure for the inverse application of the model was developed and applied to estimate the powder flow properties starting from msFBR data. The application of this procedure provided good results in terms of effective angle of internal friction and is promising for the ability of the system to explore powder flow at very low consolidation states.     

Formation of β-cylindrites under supercooled extrusion of isotactic polypropylene at low shear stress

Formation of β-cylindrites of isotactic polypropylene under various wall shear stress (σ_w), supercooled temperature of melt (T_e) and crystallization temperature (T_c) has been investigated by polarized light microscopy (PLM), wide angle X-ray diffraction (WAXD), and differential scanning calorimeter (DSC). To have better control over the thermomechanical history, instead of a reciprocating screw, the samples were prepared by extruding supercooled melt through capillary die. β-cylindrites can be observed by PLM in the extruded specimen even at a lower σ_w (0.020 MPa), and the number of β-cylindrites nuclei increases rapidly with the lowering of T_e. The nucleation density of β-cylindrites increases with the raising of wall shear stress under a given T_e of 160 °C. Furthermore, at lower supercooled temperature of melt (145 °C), the radius of β-cylindrites decreases with the increasing of σ_w, and the number of β-cylindrites nuclei almost remain invariant. At relatively higher σ_w (0.090 MPa), a saturation of β-cylindrites nuclei is observed with decreasing T_c. A modified model based on above results has been proposed to explain the effect of the original structure of quiescent supercooled melt on the formation of β-cylindrites under low shear stress.     

Application of critical state to uniaxial compaction

From the radial stress σ_R and the normal stress σ_A, measured continuously during uniaxial loading and unloading on three compactable (sodium chloride, polyethylene and tartaric acid) and two non-compactable (polypropylene and polystyrene plastics) materials, characteristic compaction profiles of (σ_A ? σ_R) versus (σ_A + σ_R) can be observed. The uniaxial loading stress pathways for both compactable and non-compactable materials validated the assumption that the Coulomb yield criterion, which is usually applicable for the shear testing of soils, can be applied to the uniaxial compression of particulate materials. In addition, the unloading stress profiles for the compactable materials produced two characteristic parameters: a normal stress value at zero shear (σ_A + σ_R)_o and a minimum shear stress value (σ_A ? σ_R)_min. Correlation of (σ_A + σ_R)_o and (σ_A ? σ_R)_min values with either the tensile strength f_c or the Vickers hardness number H_V from the resultant compacts showed a linear logarithmic relationship. No such relationship was found, however, with non-compactable materials.     

Molecular weight distribution and rheological properties of polyisobutylene solutions

The molecular weight distribution of a series of polyisobutylenes was determined using osmotic pressure measurements, gel permeation chromatography, and intrinsic viscosity. All of the polymers except for one, a blend of the highest and lowest molecular weight constituents, had similar moderate molecular weight distributions. The “extended chain length” method of calibrating the gel permeation chromatograph for polyisobutylenes was found to be effective. Steady state and transient shear stresses and normal stresses were measured on 5% decalin solutions of these polymers. The zero shear viscosity increased with the 3.3 power of molecular weight, and the zero shear normal stress coefficient (σ₁₁ ? σ₂₂)/Γ² varied with the 7.5 power. Relative elastic memory as measured by (σ₁₁ ? σ₂₂)/σ₁₂ or stress relaxation increased with increasing molecular weight (and at constant number- or weight-average molecular weight) with breadth of distribution. Stress overshoot also correlated with this tendency.     

Interactive software for calculating the principal stresses of compacted cohesive powders with the Warren-Spring equation

The Unconfined Yield Stress (σ_c) and Major Consolidation Stress (σ₁) of a cohesive powder′s compact are found by constructing two Mohr semicircles that are tangential to the Yield Loci Curve (YLC); the first passing through the origin (0,0) and the second at the consolidation conditions (σ₀,τ₀). When the YLC can be described by the Warren-Spring equation (τ/C)ⁿ = (σ + Τ)/Τ or an alternative algebraic expression, this translates into finding the solution of two pairs of simultaneous equations that set the conditions for the tangential YLC and corresponding Mohr semicircles to have the same value and slope at their respective contact points. Once the Mohr semicircle′s equation that corresponds to the consolidation conditions has been found, the Effective Angle of Internal Friction (δ) is calculated in a similar manner. The numerical calculation procedure has been automated in a freely downloadable program posted on the web as a Wolfram Project Demonstration. It allows the user to choose and adjust the values of C, T, n and σ₀, and the plot′s scales, by moving sliders on the computer screen. The program calculates and displays the corresponding values of σ_c, σ₁ and δ, and plots the YLC, two Mohr semicircles and the line that defines δ. Since a linear YLC is just a special case of the model where n = 1, the program can be used with input parameters originally obtained by linear regression. But although the program can render reasonable estimates of the principal stresses σ_c, σ₁ and δ in this case too, the physical meaning of C, and especially T, is unclear when calculated by extrapolation instead of being determined experimentally.     

Flow Properties of Highly Dispersed Powders at Very Small Consolidation Stresses

With a modified standard ring shear tester yield loci of highly dispersed, dry powders were measured at preshear normal stresses down to 32 Pa and shear stresses down to 10 Pa. At small consolidation stresses stress, σ₁, (< 500 Pa) the values obtained for the unconfined yield strength, σ_c, are proportional to the consolidation stress, σ₁.     

Synthesis and dissolution behavior of β-TCP and HA/β-TCP composite powders

Calcium phosphate powders, β-TCP and biphasic HA/β-TCP, were synthesized by calcining the powders obtained from the co-precipitation method using Ca(NO₃)₂·4H₂O and (NH₄)₂HPO₄. The effects of the initial Ca/P ratio and pH of the solution on the phase evolution and in vitro dissolution behavior of the powders in a Ringer's solution were investigated. The Ca/P ratio of the resulting powders was strongly dependent on the pH of the solution and weakly dependent on the initial Ca/P ratio. Single phase TCP powder was obtained at pH=7.4 and the initial Ca/P ratio had a little effect on the resulting Ca/P ratio. Biphasic composite powders were prepared at pH=8.0 and the Ca/P ratio of resulting powder was controllable by adjusting the initial Ca/P ratio. TCP powder showed the highest dissolution rate in the Ringer's solution and biphasic composite powder exhibited an intermediate dissolution behavior between that of HA and TCP.     

Synthesis and electrochemical properties of HT-LiCo0.8Ni0.2O2 prepared by molten salt synthesis method using 0.59LiNO3-0.41LiOH system

LiCo_0.8Ni_0.2O₂ powders were prepared by molten salt synthesis method using a 0.59LiNO₃-0.41LiOH melt. The physico-chemical properties of the HT-LiCo_0.8Ni_0.2O₂ powders were investigated by powder X-ray diffraction method, field emission scanning electron microscopy, cyclic voltammetry, and charge-discharge cycling. Uniform size nanocrystalline powder (~40 nm) could be obtained at temperature ranges of 280-480 °C. These nanocrystalline powders showed poor electrochemical properties because of presence of the low temperature phase. Pure single-phase HT-LiCo_0.8Ni_0.2O₂ powder could be obtained at 580 °C. Charge-discharge measurements indicated that this exhibited a good capacity and cyclability without further heat treatment at higher temperatures. Presented at the Int’l Symp. on Chem. Eng. (Cheju, Feb. 8-10, 2001), dedicated to Prof. H. S. Chun on the occasion of his retirement from Korea University.     

An analytical hopper design method for cohesive powders

This method overcomes two practical difficulties in the Jenike graphical solution: that of making an accurate estimation of the material effective angle of internal friction δ (which we show to be a function of flow factor, ff_s), and then of extrapolating the powder failure function FF curve to obtain a correct intersection with the hopper flow factor line.We find that FF and all the inter-related material internal friction angles of simple powders are themselves functions of specific tension T_s (tensile stress T divided by its associated steady-state normal stress σ_L), the critical value of which may be calculated from

where n and K are powder flow property constants determined experimentally; Y is the yield factor, a constant combining the other two and linking yield stress f at the free surface of the powder arch directly with tensile stress T; and ff_s is a flow factor calculated at the outlet stresses developed during hopper charging, and which assumes arch failure along the plane of the principal shear stress. This critical value of T_s then determines the maximum stable arch span B from

in which $\text{b}$ and $\text{β}$ are the suggested straight-line constants of the time-consolidated locus, and c and γ are those of the proposed tensile stress—bulk density, ?, relationship, T = c?^γ, all of which are determined experimentally.This solution fully takes into account the effect of material compressibility and hopper surcharge pressures, and ensures an answer on the conservative side. A computer programme (HOPACALC) has been written in Fortran to handle the calculations.     

Effects of interlaminar stresses on damping of 0-degree unidirectional laminated composites

The principal aim of this paper is to formulate a general model for predicting damping in composites on the basis of the concept of strain energy-weighted dissipation. In this model, the effects of interlaminar stresses on damping have been included in addition to the effects of in-plane extension/compression and in-plane shear. Validation of the model was confirmed by performing damping measurements on 0° unidirectional composite beams with varying length and thickness. The results of theoretical predictions of damping in laminated composites were found to compare favorably with experimental data. The transverse shear (σ_xz) reveals to have a considerable effect on the damping mechanisms in 0° unidirectional polymer composites. However, the other interlaminar stresses (σ_yz, σ_z) were shown to have little influence on damping in composite beam.