Effect of γ − α′ phase transformation on plastic adaptation to cyclic loads at cryogenic temperatures

Metastable, type FCC metals and alloys are often applied at extremely low temperatures because of their excellent ductility over the whole temperature range practically down to the absolute zero. These materials (like stainless steels) are frequently characterised by the low stacking fault energy and undergo at low temperatures the plastic strain induced transformation from the parent phase “γ” to the secondary phase “α′”. The phase transformation process consists in creation of two-phase continuum, where the parent phase coexists with the inclusions of secondary phase in thermodynamic equilibrium. The evolution of material micro-structure induces strain hardening related to interaction of dislocations with the inclusions and to increase of equivalent tangent stiffness as a result of evolving proportions of both phases, each characterised by different stiffness. The corresponding hardening model is based on micromechanics and on the Hill concept (1965) supplemented by Mori and Tanaka (1973) homogenisation scheme. Identification of parameters of the constitutive model has been carried out for 304L and 316L stainless steels, based on the available experimental data. The model has been used to describe phase transformation in rectangular beams, circular rods and thin-walled shells subjected to cyclic loads at cryogenic temperatures. Moreover, non-proportional loading paths were studied. A new feature of structures made of metastable materials has been observed. As soon as the γ ? α′ phase transformation begins, the evolution of material micro-structure accelerates the process of adaptation of structural member to cyclic loads and enhances therefore its fatigue life when compared to classical elastic–plastic structures.     

Functionally graded structural members obtained via the low temperature strain induced phase transformation

The notion of functionally graded materials (FGM) covers all domains of discrete and smooth gradation of material microstructure designed in order to obtain macroscopic features suitable for a given application. A special class of multi-phase materials with graded microstructure can be obtained at cryogenic temperatures as a result of smooth transition from the parent phase to the secondary phase. The required continuously graded material features are obtained at low temperatures via the mechanism of controlled strain induced phase transformation from the purely austenitic to the martensitic lattice (γ → α′). Several families of ductile materials are known to behave in a metastable way when strained at very low temperatures. Among them the austenitic stainless steels are extensively used to construct components of the superconducting magnets, cryogenic transfer lines and other structural members loaded in cryogenic conditions. The constitutive model used to describe mathematically the plastic strain induced phase transformation at low temperatures involves strain hardening where two fundamental effects play an important role: interaction of dislocations with the martensite inclusions and increase in material tangent stiffness due to the mixture of harder martensite with softer austenite. The interaction of dislocations with the martensite inclusions is reflected by the hardening modulus that depends on the volume fraction of martensite. Here, a linear approximation, based on the micro-mechanics analysis, is used. On the other hand, evaluation of the material tangent stiffness of two-phase continuum is based on the classical homogenization scheme and takes into account the local tangent moduli of the components, as postulated by Hill [Hill, R., 1965. A self consistent mechanics of composite materials. J. Mech. Phys. Solids 13, 213–222]. In the present paper, the Mori–Tanaka homogenisation scheme is applied. Both effects contribute to strong nonlinear hardening that occurs as soon as the phase transformation process begins. The material model is suitable for a wide range of temperatures, however the best results are obtained at very low temperatures, where the linearized kinetic law of phase transformation is valid [Garion, C., Skoczeń B., 2002. Modeling of plastic strain induced martensitic transformation for cryogenic applications. J. Appl. Mech. 69 (6), 755–762]. As the application field the structural members in the form of rods (cylinders) of circular cross-section, used as parts of the carrying structures, are analyzed. The required graded microstructure of the material is obtained by imposing torsion at cryogenic temperatures. Both the intensity of the phase transformation and the depth of the transformed zone is obtained by suitable kinematic control (angle of twist). The closed form solutions for the stress state and torque as a function of the angle of twist are shown.     

Reconcile the perfectly elastoplastic model to simulate the cyclic behavior and ratcheting

Perfectly elastoplastic constitutive model is modified through a smoothing factor introduced by Liu [Liu, C.-S., 2003. Smoothing elastoplastic stress–strain curves obtained by a critical modification of conventional models. Int. J. Solids Struct. 40, 2121–2145]. The new model allows plasticity to happen in a non-zero-measure yield volume in stress space, rather than that of conventional zero-measure yield surface, and within the yield volume the plastic modulus is varying continuously. It endows a specific strain-hardening rule of flow stress and is able to describe the phenomena of strain hardening, cyclic hardening, the Bauschinger effect, mean-stress relaxation, strain ratcheting, out-of-phase hardening, as well as erasure-of-memory. In order to suppress the over prediction of ratcheting we consider a scalar function of smoothing factor, which can simulate the saturation behavior of uniaxial/multiaxial strain ratcheting. These effects are demonstrated through numerical examples. The existence of stress equilibrium point and limiting surface is a natural result without requiring an extra design. Moreover, the non-linear constitutive equations can be converted into a linear system for augmented stress in the Minkowski space, of which the symmetry group is a proper orthochronous Lorentz group SO_o(5, 1). The augmented stress is a time-like vector, moving on hyperboloids inside the cone. When taking the Prager kinematic hardening rule into account we can simulate some cyclic behaviors of SAE 4340 and grade 60 steels within a certain accuracy through the use of only three material constants and a fixed smoothing factor. To simulate the ratcheting behaviors of SS304 stainless steel we allow the smoothing factor to be an exponential decaying function of λ.     

Strain gradient plasticity analysis of transformation induced plasticity in multiphase steels

“To what extent do plastic strain gradients affect the strengthening resulting from the transformation of small metastable inclusions into hard inclusions within a plastically deforming matrix?” is the central question addressed here. Though general in the approach, the focus is on the behavior of TRIP-assisted multiphase steels. A two-dimensional embedded cell model of a simplified microstructure composed of a single metastable austenitic inclusion surrounded by a soft ferritic matrix is considered. The cell is inserted in a large homogenized medium. The transformation of a fraction of the austenite into a hard martensite plate is simulated, accounting for a transformation strain, and leading to complex elastic and plastic accommodation. The size of a transforming plate in real multiphase steels is typically between 0.1 and 2 μm, a range of size in which plastic strain gradient effects are expected to play a major role. The single parameter version of the Fleck–Hutchinson strain gradient plasticity theory is used to describe the plasticity in the austenite, ferrite and martensite phases. The higher order boundary conditions imposed on the plastic flow have a large impact on the predicted strengthening. Using realistic values of the intrinsic length parameter setting the scale at which the gradients effects have an influence leads to a noticeable increase of the strengthening on top of the increase due to the transformation of a volume fraction of the retained austenite. The geometrical parameters such as the volume fraction of retained austenite and of the transforming zone also bring significant strengthening. Strain gradient effects also significantly affect the stress state inside the martensite plate during and after transformation with a potential impact on the damage resistance of these steels.     

A thermomechanical constitutive model for phase transformations in silicon under pressure and contact loading conditions

Most of the technologically relevant abrasive machining techniques for silicon (Si) such as lapping, sawing and grinding are based on the interaction of the silicon surface with a hard particle or asperity. It has been long established that the governing deformation mechanism for Si under such contact loading conditions is stress induced phase transformation. The present work introduces a novel phenomenological constitutive model for phase transformations of silicon set up in a thermomechanical framework of broad applicability. Taking into account experimental observations as well as first principle and molecular dynamics calculations, it captures both the cd-Si → β-Si transition upon compression and the β-Si → a-Si transition upon rapid decompression, which are most relevant for indenter loading. The model was numerically implemented in analogy to incremental plasticity and successfully applied for finite-element (FE) simulations of nanoindentation.     

Failure in notched tension bars due to high-temperature creep: Interaction between nucleation controlled cavity growth and continuum cavity growth

For axi-symmetrically notched tension bars [Dyson, B.F., Loveday, M.S., 1981, Creep Fracture in Nimonic 80A under Tri-axial Tensile Stressing, In: Ponter A.R.S., Hayhurst, D.R. (Eds.), Creep in Structures, Springer-Verlag, Berlin, pp. 406–421] show two types of damage propagation are shown: for low stress, failure propagates from the outside notch surface to the centre-line; and for high stress, failure propagates from the centre-line to the outside notch surface. The objectives of the paper are to: identify the physics of the processes controlling global failure modes; and, describe the global behaviour using physics-based constitutive equations.Two sets of constitutive equations are used to model the softening which takes place in tertiary creep of Nimonic 80A at 750 °C. Softening by multiplication of mobile dislocations is firstly combined, for low stress, with softening due to nucleation controlled creep constrained cavity growth; and secondly combined, for high stress, with softening due to continuum void growth. The Continuum Damage Mechanics, CDM, Finite Element Solver DAMAGE XX has been used to study notch creep fracture. Low stress notch behaviour is accurately predicted provided that the constitutive equations take account of the effect of stress level on creep ductility. High stress notch behaviour is accurately predicted from a normalized inverse cavity spacing d/2? = 6, and an initial normalized cavity radius r_hi/? = 3.16 × 10^?3, where 2? is the cavity spacing, and d is the grain size; however, the constants in the strain rate equation required recalibration against high stress notch data. A void nucleation mechanism is postulated for high stress behaviour which involves decohesion where slip bands intersect second phase grain boundary particles. Both equation sets accurately predict experimentally observed global failure modes.     

Phenomenological study of parabolic and spherical indentation of elastic-ideally plastic material

A phenomenological study of parabolic and spherical indentation of elastic ideally plastic materials was carried out by using precise results of finite elements calculations. The study shows that no “pseudo-Hertzian” regime occurs during spherical indentation. As soon as the yield stress of the indented material is exceeded, a deviation from the, purely elastic Hertzian contact behaviour is found. Two elastic–plastic regimes and two plastic regimes are observed for materials of very large Young modulus to Yield stress ratio, E/σ_y. The first elastic–plastic regime corresponds to a strong evolution of the indented plastic zone. The first plastic regime corresponds to the commonly called “fully plastic regime”, in which the average indentation pressure is constant and equal to about three times the yield stress of the indented material. In this regime, the contact depth to penetration depth ratio tends toward a constant value, i.e. h_c/h = 1.47. h_c/h is only constant for very low values of yield strain (σ_y/E lower than 5 × 10^?6) when aE¹/Rσ_y is higher than 10,000. The second plastic regime corresponds to a decrease in the average indentation pressure and to a steeper increase in the pile-up. For materials with very large E/σ_y ratio, the second plastic regime appears when the value of the non-dimensional contact radius a/R is lower than 0.01. In the case of spherical and parabolic indentation, results show that the first plastic regime exists only for elastic-ideally plastic materials having an E/σ_y ratio higher than approximately 2.000.     

An experimental and numerical study of the localization behavior of tantalum and stainless steel

In general, the shear localization process involves initiation and growth. Initiation is expected to be a stochastic process in material space where anisotropy in the elastic–plastic behavior of single crystals and inter-crystalline interactions serve to form natural perturbations to the material’s local stability. A hat-shaped sample geometry was used to study shear localization growth. It is an axi-symmetric sample with an upper “hat” portion and a lower “brim” portion with the shear zone located between the hat and brim. The shear zone length is 870–890 μm with deformation imposed through a split-Hopkinson pressure bar system at maximum top-to-bottom velocity in the range of 8–25 m/s. We present experimental results of the deformation response of tantalum and 316L stainless steel samples. The tantalum samples did not form shear bands but the stainless steel sample formed a late stage shear band. We have also modeled these experiments using both conductive and adiabatic continuum models. An anisotropic elasto-viscoplastic constitutive model with damage evolution was used within the finite element code EPIC. A Mie-Gruneisen equation of state and the rate and temperature sensitive MTS flow stress model together with a Gurson flow surface were employed. The models performed well in predicting the experimental data. The numerical results for tantalum suggested a maximum equivalent strain rate on the order of 7 × 10⁴ s⁻¹ in the gage section for an imposed top surface displacement rate of 17.5 m/s. The models also suggested that for an initial temperature of 298 K a temperature in the neighborhood of 900 K was reached within the shear section. The numerical results for stainless steel suggest that melting temperature was reached throughout the shear band shortly after peak load. Due to sample geometry, the stress state in the shear zone was not pure shear; a significant normal stress relative to the shear zone basis line was developed.     

Thermo-mechanical behaviour of TRIP 1000 steel sheets subjected to low velocity perforation by conical projectiles at different temperatures

This paper presents and analyzes the behaviour of TRIP 1000 steel sheets subjected to low velocity perforation by conical projectiles. The relevance of this material resides in the potential transformation of retained austenite to martensite during impact loading. This process leads to an increase in strength and ductility of the material. However, this transformation takes place only under certain loading conditions strongly dependent on the initial temperature and deformation rate. In order to study the material behaviour under impact loading, perforation tests have been performed using a drop weight tower. Experiments were carried out at two different initial temperatures T₀ = 213 K and T₀ = 288 K, and within the range of impact velocities 2.5 m/s ? V₀ ? 4.5 m/s. The experimental setup enabled the measuring of impact velocity, residual velocity, load-time history and failure mode. In addition, dry and lubricated contacts between the striker and the plate have been investigated. Finally, by using X-ray diffraction it has been shown that no martensitic transformation takes place during the perforation process. The causes involving the none-appearance of martensite are examined.     

Simple constitutive modelling of nonlinear viscoelasticity under general extension

We study the behaviour of a single integral constitutive equation, capable of providing analytic expressions for the viscoelastic stress in extensional flows of a variety of deformation histories and geometries, ranging from uniaxial to equibiaxial. It is based on the use of a stress damping function, with a power-law dependence on the elongation, λ: h(λ) = 1/λⁿ. The parameter n (0 ≤ n ≤ 2) signifies the nonlinear viscoelastic character of the material and, therefore, is an inverse measure of network connectivity strength of the underlying microstructure. This renders the constitutive approach applicable to incompressible polymers of a variable degree of branching, strain hardening and stress thinning behavior. Methods of connecting n with the macromolecular architecture and the alignment strength of the flow are also explored.     

Evaluation of subgrid-scale models in large-eddy simulation of flow past a two-dimensional block

Large-eddy simulations of flow past a two-dimensional (2D) block were performed to evaluate four subgrid-scale (SGS) models: (i) the traditional Smagorinsky model, (ii) the Lagrangian dynamic model, (iii) the Lagrangian scale-dependent dynamic model, and (iv) the modulated gradient model. An immersed boundary method was employed to simulate the 2D block boundaries on a uniform Cartesian grid. The sensitivity of the simulation results to grid refinement was investigated by using four different grid resolutions. The velocity streamlines and the vertical profiles of the mean velocities and variances were compared with experimental results. The modulated gradient model shows the best overall agreement with the experimental results among the four SGS models. In particular, the flow recirculation, the reattachment position and the vertical profiles are accurately reproduced with a relative coarse grid resolution of (N_x × N_y × N_z=) 160 × 40 × 160 (n_x × n_z = 13 × 16 covering the block). Besides the modulated gradient model, the Lagrangian scale-dependent dynamic model is also able to give reasonable prediction of the flow statistics with some discrepancies compared with the experimental results. Relatively poor performance by the Lagrangian dynamic model and the Smagorinsky model is observed, with simulated recirculating patterns that differ from the measured ones. Analysis of the turbulence kinetic energy (TKE) budget in this flow shows evidence of a strong production of TKE in the shear layer that forms as the flow is deflected around the block.     

Experimental quantification of the plastic blunting process for stage II fatigue crack growth in one-phase metallic materials

The plastic blunting process during stage II fatigue crack growth was studied in pure polycrystalline Ni to investigate effects of strain localization and inelastic behavior on the kinematics of crack advance. Correlations were obtained between strain fields ahead of a fatigue crack, crack advance per cycle and crack growth kinetics. Strain fields were quantified using a combination of in situ loading experiments, scanning electron microscopy and digital image correlation for 8 < ΔK < 20 MPa m^1/2 and a fixed load ratio of 0.1. Results indicate that strain localized along a dominant deformation band, which was usually crystallographic and carried mostly pure shear for large loads and was of mixed character for lower loads. Instances of double deformation bands were observed, with bands acting either in a simultaneous or alternating fashion. It was found that the area integral of the opening strain for values larger than a given threshold, an “integrated” strain, had a power-law relationship with ΔK, with the exponent approximately equal to the Paris exponent (m). Therefore, the crack growth rate was proportional to the integrated strain. An analysis based on this correlation and the presence of dominant shear bands indicated that the integrated strain is related to the accumulated displacement in the band. This, in turn, is proportional to the product of the cyclic plastic zone radius and the average shear strain ahead of the tip, which represents a basic length scale for plastic blunting. Assumptions on the load dependence of these quantities, based on their observed spatial variation, allowed estimating $m=2\left(1+\frac{1}{1+n^{\prime }}\right)$, where n′ is the cyclic hardening exponent (0 < n′ < 1). This gives 3 < m < 4, which accounts for about 50% of the observed values of m between 1.5 and 6 for a wide variety of metallic materials.     

Optimization and evaluation of fish oil microcapsules

Fish oil microcapsules were prepared using two natural polysaccharides, alginate and chitosan, as the wall materials. A response surface methodology (RSM) was used to optimize the conditions for fish oil encapsulation efficiency (FOEE). The FOEE was investigated with respect to three key-variables in the RSM: ratio of inner oil phase to aqueous phase (X₁, w/w); concentration of the aqueous phase (X₂, wt%); and ratio of the aqueous phase to outer oil phase (X₃, v/v). The optimal formulation obtained from the RSM model, i.e., 2.7:1 (X₁), 1.6 wt% (X₂), and 11.5:1 (X₃), gave a FOEE of 28%. The model was validated and the fish oil microcapsules prepared under the optimized conditions were characterized in terms of particle size, polydispersity index (PDI), zeta potential, surface morphology, and in vitro release. The average droplet size, PDI, and zeta potential were 915 nm, 0.038, and +5.2 mV, respectively. The fish oil microcapsules were highly uniform microspheres, and had an accumulative release rate of 77.7% in 270 min in a gastrointestinal model, indicating their potential as an alternative carrier for the controlled release of fish oil. In conclusion, formulating optimal microencapsulation conditions by the RSM can be applied to the microencapsulation of various oil-soluble nutrients for food applications.     

Asymptotic crack-tip fields for dynamic crack growth in compressive power-law hardening materials

The effect of material compressibility on the stress and strain fields for a mode-I crack propagating steadily in a power-law hardening material is investigated under plane strain conditions. The plastic deformation of materials is characterized by the J₂ flow theory within the framework of isotropic hardening and infinitesimal displacement gradient. The asymptotic solutions developed by the present authors [Zhu, X.K., Hwang K.C., 2002. Dynamic crack-tip field for tensile cracks propagating in power-law hardening materials. International Journal of Fracture 115, 323–342] for incompressible hardening materials are extended in this work to the compressible hardening materials. The results show that all stresses, strains, and particle velocities in the asymptotic fields are fully continuous and bounded without elastic unloading near the dynamic crack tip. The stress field contains two free parameters σ_eq0 and s₃₃₀ that cannot be determined in the asymptotic analysis, and can be determined from the full-field solutions. For the given values of σ_eq0 and s₃₃₀, all field quantities around the crack tip are determined through numerical integration, and then the effects of the hardening exponent n, the Poisson ratio ν, and the crack growth speed M on the asymptotic fields are studied. The approximate behaviors of the proposed solutions are discussed in the limit of ν → 0.5 or n → ∞.     

Hydrodynamic forces on a rotating sphere

The wake dynamics of a rotating sphere with prescribed rotation axis angles are quantitatively analysed by carrying out numerical simulations at Reynolds numbers of Re = 100, 250 and 300, non-dimensional rotational rates Ω¹ = 0–1 and rotation axis angles α = 0, π/6, π/3 and π/2 measured from the free stream axis. These parameters are the same as those in an earlier study (Poon et al., 2010, Int. J. Heat Fluid Flow) where the instantaneous flow structures were discussed qualitatively. This study extends the findings of the earlier study by employing phase diagrams (C_Lx, C_Ly) and (C_D, C_L) to provide a quantitative analysis of the time-dependent behaviour of the flow structures. At Re = 300 and Ω¹ = 0.05, the phase diagrams (C_Lx, C_Ly) show ‘saw tooth’ patterns for both α = 0 and π/6. The ‘saw tooth’ pattern indicates that the flow structures comprise a higher frequency oscillation component at a Reynolds number of 300 which is not observed until Re ≈ 800 for a stationary sphere. This ‘saw tooth’ pattern disappears as Ω¹ increases. The employment of the phase diagrams also reveals that different flow structures induce different oscillation amplitudes on both lateral force coefficients. With the exception of the vortices formed from a shear layer instability, all other flow regimes show larger fluctuations in C_L than C_D.     

Revisiting the perforation of ductile plates by sharp-nosed rigid projectiles

The process of ductile plate perforation by sharp-nosed rigid projectiles is further examined in this work through 2D numerical simulations. We highlight various features concerning the effective resisting stress (σ_r) which a finite thickness plate, with a flow stress of Y_t, exerts on the projectile during perforation. In particular, we show that the normalized resisting stress (σ_r/Y_t) can be represented as a unique function of the normalized thickness of the plate (H/D, where H is plate thickness and D is projectile diameter), for a large range of normalized thicknesses. Our simulations for very thin target plates show that the penetration process is achieved through the well-known dishing mechanism, where the target material is pushed forward by the projectile’s nose. An important observation, which emerges from our simulations, is that the transition between the dishing and the hole enlargement mechanisms takes place at a normalized thickness of about H/D = 1/3. We also find that the normalized resistive stress for intermediate plate thicknesses, 1/3 < H/D < 1.0, is relatively constant at a value of σ_r/Y_t = 2.0. This range of thicknesses conforms to a state of quasi plane stress in the plates. For thicker plates (H/D > 1) the σ_r/Y_t ratio increases monotonically to values which represent the resistance to penetration of semi-infinite targets, where the stress state is characterized by plane strain conditions. Using a simple model, which is based on energy conservation, we can predict the values of the ballistic limit velocities for many projectile/target combinations, provided the perforation is done through the ductile hole enlargement mechanism. Good agreement is demonstrated between predictions from our model and experimental data from different sources, strongly enhancing the confidence in both the validity and usefulness of our model.     

Flow evolution of a turbulent submerged two-dimensional rectangular free jet of air. Average Particle Image Velocimetry (PIV) visualizations and measurements

The paper presents average flow visualizations and measurements, obtained with the Particle Image Velocimetry (PIV) technique, of a submerged rectangular free jet of air in the range of Reynolds numbers from Re = 35,300 to Re = 2200, where the Reynolds number is defined according to the hydraulic diameter of a rectangular slot of height H. According to the literature, just after the exit of the jet there is a zone of flow, called zone of flow establishment, containing the region of mixing fluid, at the border with the stagnant fluid, and the potential core, where velocity on the centerline maintains a value almost equal to the exit one. After this zone is present the zone of established flow or fully developed region. The goal of the paper is to show, with average PIV visualizations and measurements, that, before the zone of flow establishment is present a region of flow, never mentioned by the literature and called undisturbed region of flow, with a length, L_U, which decreases with the increase of the Reynolds number. The main characteristics of the undisturbed region is the fact that the velocity profile maintains almost equal to the exit one, and can also be identified by a constant height of the average PIV visualizations, with length, L_CH, or by a constant turbulence on the centerline, with length L_CT. The average PIV velocity and turbulence measurements are compared to those performed with the Hot Film Anemometry (HFA) technique. The average PIV visualizations show that the region of constant height has a length L_CH which increases from L_CH = H at Re = 35,300 to L_CH = 4–5H at Re = 2200. The PIV measurements on the centerline of the jet show that turbulence remains constant at the level of the exit for a length, L_CT, which increases from L_CT = H at Re = 35,300 to L_CT = 4–5H at Re = 2200. The PIV measurements show that velocity remains constant at the exit level for a length, L_U, which increases from L_U = H at Re = 35,300 to L_U = 6H at Re = 2200 and is called undisturbed region of flow. In turbulent flow the length L_U is almost equal to the lengths of the regions of constant height, L_CH, and constant turbulence, L_CT. In laminar flow, Re = 2200, the length of the undisturbed region of flow, L_U, is greater than the lengths of the regions of constant height and turbulence, L_CT = L_CH = 4–5H. The average PIV and HFA velocity measurements confirm that the length of potential core, L_P, increases from L_P = 4–5H at Re = 35,300 to L_P = 7–8H at Re = 2200, and are compared to the previous experimental and theoretical results of the literature in the zone of mixing fluid and in the fully developed region with a good agreement.     

Spatiotemporal variations of PM2.5 and PM10 concentrations between 31 Chinese cities and their relationships with SO2, NO2, CO and O3

The variations of mass concentrations of PM_2.5, PM₁₀, SO₂, NO₂, CO, and O₃ in 31 Chinese provincial capital cities were analyzed based on data from 286 monitoring sites obtained between March 22, 2013 and March 31, 2014. By comparing the pollutant concentrations over this length of time, the characteristics of the monthly variations of mass concentrations of air pollutants were determined. We used the Pearson correlation coefficient to establish the relationship between PM_2.5, PM₁₀, and the gas pollutants. The results revealed significant differences in the concentration levels of air pollutants and in the variations between the different cities. The Pearson correlation coefficients between PMs and NO₂ and SO₂ were either high or moderate (PM_2.5 with NO₂: r = 0.256–0.688, mean r = 0.498; PM₁₀ with NO₂: r = 0.169–0.713, mean r = 0.493; PM_2.5 with SO₂: r = 0.232–0.693, mean r = 0.449; PM₁₀ with SO₂: r = 0.131–0.669, mean r = 0.403). The correlation between PMs and CO was diverse (PM_2.5: r = 0.156–0.721, mean r = 0.437; PM₁₀: r = 0.06–0.67, mean r = 0.380). The correlation between PMs and O₃ was either weak or uncorrelated (PM_2.5: r = −0.35 to 0.089, mean r = −0.164; PM₁₀: r = −0.279 to 0.078, mean r = −0.127), except in Haikou (PM_2.5: r = 0.500; PM₁₀: r = 0.509).