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Abstract

Shear and extensional properties of high-density polyethylene (HDPE) and poly(butylene succinate) (PBS) blends were measured using capillary rheometers to investigate the influences of processing conditions and mixing ratios on flow behaviors, and the predictive abilities of extensional master curve, back propagation neural network model, and constitutive equation were analyzed for the extensional flow. The results showed that the shear flow of blend melts obeyed the power law, and the influence of PBS content on shear-thinning behavior was obvious. The extensional viscosity of the blends decreased with extensional strain rate, extrusion rate, and PBS content increasing. The extensional master curve was applied for HDPE/PBS blends, and scaling factor reduced with the increasing extrusion rate. In addition, the approach of back propagation neural network model was an effective approach to predict extensional viscosity for the purposes of polymer processing control and monitoring.

Keywords

High-density polyethylene poly(butylene succinate)flow viscosity prediction

Introduction

Shear and extensional viscosities, two important processing parameters, can control both the processing properties of polymeric products.^1

–5 Studying the shear and extensional viscosities under different conditions can be helpful to reveal flow mechanisms and provide some guidelines for optimizing processing technology. Capillary rheometer is widely used in research and industry to determine the flow properties of polymeric materials over very wide ranges of shear rates. A melt spinning technique, close to isothermal uniaxial tension, is performed using the capillary rheometer equipped with a specific extensional device. It is similar to industrial polymer processes like fiber spinning and film casting. It is convenient to obtain the extensional viscosity data using the melt spinning technique, and the data show good reproducibility.^6
–8 Therefore, this testing method is widely applied to measure the extensional properties of polymeric materials.

In general, the extensional rates are high to obtain the extensional viscosities at high extensional strain rates and large extrusion rates. However, polymer filaments adhere to a traction wheel, resulting in experimental failure. Thus, it is important to predict the extensional viscosities at high strain rates and large extrusion rates. Previously, three approaches based on extensional master curve, back propagation neural network model, and constitutive equation are applied to predict some processing parameters, such as extensional viscosity,^9,10 extrusion pressure,¹¹ and injection pressure.¹² The extensional master curve depicts the response of extensional force to extensional ratio, and the response depends on both processing conditions and measured instrument in the spinning test of pure polymer melts.⁸ The back propagation neural network model, as an artificial neural network model, can mimic and process known information to predict unknown information just like the brain of higher animals.¹³ Moreover, the constitutive equations based on macromolecular structures have been conducted to predict the extensional viscosity directly.^9,10 Thus, it is necessary to search an effective approach to predict the extensional viscosity by comparison between the predictive abilities of three approaches.

High-density polyethylene (HDPE) is a common thermoplastic resin with favorable mechanical and processing properties as well as low price. However, HDPE made from nonrenewable fossil fuels is consumed and discarded into nature environment, resulting in nondegradable wastes. Poly(butylenes succinate) (PBS), as a promising biodegradable material, is a linear polyester.^14,15 Compared with HDPE, PBS is expensive, and some properties of PBS (e.g. viscosity, melting point, and tensile strength) restrict its applications. Therefore, blending of PBS with HDPE is an economical and effective way to lower the cost and enhance the degradability of HDPE.^16
–18 In this article, the shear viscosity of the HDPE/PBS blends was measured with the capillary rheometry, the extensional viscosity was obtained using melt spinning technique, and the effects of processing conditions (e.g. shear rate, temperature, extensional strain rate, and extrusion rate) and mixing ratios on shear and extensional flows were discussed for HDPE/PBS blends. Furthermore, the predictive abilities of extensional viscosity based on the extensional master curve, the back propagation neural network model, and the constitutive equation were analyzed to provide useful basis for the online processing of HDPE/PBS blends.

Experimental

Raw materials

HDPE, the resin with the trade mark of DGDA-6094, was supplied by Guangzhou Petrochemical Works (Guangdong province, People’s Republic of China), its density in the solid state was 0.95 g cm⁻³ and its melt flow rate was 1 g/10 min. PBS, the resin with the trade mark of Bionolle 1020, was supplied by Showa Denko Co. (Tokyo, Japan), its density of the solid state was 1.26 g cm⁻³, and its melt flow rate was 25 g/10 min.

Processing of blends

HDPE and PBS resins were blended with a twin-screw extruder at the temperatures varying from 140 to 160°C, the diameter (D) and length to diameter (L/D) ratio of the screw were 22 mm and 42, respectively. Then, the extrudates of HDPE/PBS blends were cooled by water and then granulated. The weight fractions of PBS (ϕ) were 0, 20, 40, 60, and 100%. Finally, HDPE/PBS blends particles were dried for 4 h at 75°C before rheological tests.

Instrument and methodology

The shear flow test was performed on a Rheograph 25 of constant rate-type capillary rheometer, which was manufactured by Gottfertt Co. (Germany). The test temperatures ranged from 160 to 200°C and the apparent shear rates at channel wall $({\dot{γ}}_{a})$ varied from 20 to 3000 s⁻¹. The reservoir diameter and the die angle were 15 mm and 180°, respectively. The diameter (D) of the capillary was 1 mm, and the L/D ratios were 10, 20, and 30.

The wall shear stress (τ_w) is expressed by the following equation:

τ_{w} = \frac{(Δ P - Δ P_{e n t}) D}{4 L}

where ΔP is the total pressure drop in the capillary extrusion of polymer melts, ΔP_ent is the entry pressure drop which is determined by the Bagley’s plot method.

The apparent shear rate at channel wall is defined asusing the following equation:

{\dot{γ}}_{a} = \frac{32 Q}{π D^{3}}

where Q is the volumetric flow rate. According to the correction method of Rabinowitsch, the shear rate (

\dot{γ}

) at channel wall for the shear-thinning fluid is given by:

\dot{γ} = \frac{1 + 3 n}{4 n} {\dot{γ}}_{a}

Where n is the non-Newtonian index. Thus, the true shear viscosity (η_s) can be written as:

η_{s} = \frac{τ_{w}}{\dot{γ}}

The extensional flow test was performed using the Rheologic 5000 of constant rate-type capillary rheometer, which was made by Ceast Co. (Spain). In the test, the polymer melts, extruded through the capillary die, were stretched uniaxially using the melt spinning technique to measure the melt extensional properties, and the extensional force (F) measured was plotted as a function of extensional rate (V). For all the measurements, the reservoir diameter and the die angle were 15 mm and 180°, respectively. The values of D and the L/D ratio of the capillary were 1 mm and 30, respectively. Moreover, the test temperature and the distance (H) of the spinning were 180°C and 260 mm, respectively. The extrusion rates (v₀) in the capillary were 11.25, 22.5, 33.75, and 45 mm s⁻¹, and the extensional acceleration rate was 5 mm s⁻². According to the expressions derived by Laun and Schuch,¹⁹ the extensional strain rate ( $\dot{ε}$ ) and the extensional stress (σ) can be written as:

\dot{ε} = \frac{v_{0} V ln (V)}{H}

σ = \frac{4 V F}{π D^{2}}

where V is the extensional ratio, which is the ratio between the extensional rate and the extrusion rate. Thus, the calculation equation of the melt extension viscosity is given by

η_{e} = \frac{σ}{\dot{ε}} = \frac{4 F H}{π v_{0} D^{2} ln (V)}

Results and discussion

Shear flow property

Dependence of melt shear viscosity on shear rate and temperature

Figure 1(a) shows the dependence of the shear viscosity on the shear rate for the blends at 180°C. Figure 1(b) shows the dependence of the shear viscosity on the shear rate with the PBS content of 40%. It can be seen from Figure 1(a) and (b) that $lg η_{s}$ of the blends presents a linear decline with increasing $lg \dot{γ}$ , and the relationship between the shear viscosity and the shear rate accords to the power law. In general, with a rise of the shear rate, the macromolecules deviate from the equilibrium conformations, the interaction force among the macromolecules reduces, and the creation rate is less than the destruction one for the network junctions of polymer melts. Thus, the segment movement is easy, and the shear viscosity of blend melts reduces accordingly. It can be also found from Figure 1(a) that HDPE/PBS blend melts are pseudoplastic, PBS melt shows almost Newtonian behavior, and the sensitivity of the shear viscosity to the shear rate decreases with increase in PBS content. In other words, the shear-thinning behaviors of the blends weaken with the addition of PBS.

Figure 1.

Dependence of η_s on $\dot{γ}$ of HDPE/PBS blend melts (a) T = 180°C and (b) $ϕ = 40 %$ . HDPE: high-density polyethylene; PBS: poly(butylene succinate).

Dependence of melt shear viscosity on temperature and mass fraction of PBS

The dependence of the shear viscosity of the polymer melts on the temperature with PBS content of 40% is shown in Figure 2. It can be seen that $lg η_{s}$ is a linear function of 1/T (T is the absolute temperature in kelvin) approximately at fixed apparent shear rate. This indicates that the dependence of the shear viscosity on the temperature agrees with Arrhenius equation for blend melts. Moreover, the sensitivity of shear viscosity to temperature decreases with increase in shear rate. Figure 3 shows the relationship between the shear viscosity and the mass fraction of PBS at different apparent shear rates and temperature of 180°C. As shown in Figure 3, when the apparent shear rate is low and the PBS content is smaller than 40%, the shear viscosity of HDPE/PBS blends decreases quickly with increasing PBS content, and then with the further increase of PBS content, the shear viscosity reduces approximately linearly, and the extent of the reduction slows down. Furthermore, at a relative high shear rates ( ${\dot{γ}}_{a} > 800 s^{- 1}$ ), the variation of the shear viscosity is not obvious with the addition of PBS for the blend melts.

Figure 2.

Dependence of η_s on T of HDPE/PBS blend melts $(ϕ = 40 %)$ .

Figure 3.

Relationship between η_s and $ϕ$ of HDPE/PBS blend melts.

Extensional flow property

Dependence of extensional viscosity on extensional strain rate

Figure 4(a) to (c) shows the dependence of the extensional viscosity of blend melts on extensional strain rate at different extrusion rates. It can be found from Figure 4(a) to (c) that $lg η_{e}$ of blend melts reduces nonlinearly with increasing $lg \dot{ε}$ , when the extensional strain rate reaches to 3 s⁻¹ and the reduction tendency of $lg η_{e}$ slows down. In general, the macromolecules deviate from equilibrium conformations owing to the extensional deformation, and the destruction rate is more than the creation one for the network junctions. Thus, the extensional viscosity of polymer melts reduces accordingly. However, the macromolecules can orientate along the extensional direction simultaneously, and the parallel alignment of molecular chains intensifies their interactions.²⁰ So, the decreasing tendency of the extensional viscosity for the blends slows down with the further increase of the extensional strain rate.

Figure 4.

Dependence of η_e on $\dot{ε}$ of HDPE/PBS blend melts (a) $ϕ = 0$ (b) $ϕ = 20 %$ , and (c) $ϕ = 40 %$ .

Dependence of extensional viscosity on extrusion rate and PBS content

It can be also seen from Figure 4 (a) to (c) that the extensional viscosity reduces with increasing extrusion rate. On one hand, with the extrusion rate increasing, the extensional ratio of the polymer melts reduces at the same extensional strain rate, and the further extension of molecular chains is easy. On the other hand, the extensional distance is constant. With the extrusion rate increasing, the extensional time shortens and the cooling time of polymer melts decreases. Therefore, the extensional viscosity of the blends decreases. Figure 5 shows the dependence of the extensional viscosity of blend melts on PBS content at the extrusion rate of 22.5 mm s⁻¹. It can be seen from Figure 5 that the extensional viscosity reduces with increasing PBS content. At the same processing condition, the material composition is important for the extensional viscosity. Because of the low viscosity of PBS melt, the addition of PBS melt makes a lubrication effect for HDPE melt, and the flow resistance of HDPE melt decreases, resulting in the decrease of the extensional viscosity.

Figure 5.

Relationship between η_e and $ϕ$ of HDPE/PBS blend melts at extrusion rate of 22.5 mm s⁻¹.

Analysis of extensional master curve

Based on a reference curve, the response of extensional force to extensional ratio can be described by a simple scaling law, and the extensional curves (extensional force versus extensional ratio) of different extrusion rates are rescaled to get the extensional master curve by a scaling factor ( $ξ$ ).⁸ The scaling factor depends only on extrusion rate and not on extensional stress in the test.²¹ The scaling law can be expressed as:

ξ F_{i} (V_{i}) = F_{r} (\frac{V_{i}}{ξ})

where the subscript i indicates a rescaled curve, the subscript r indicates a reference curve. Different extensional curves of HDPE/PBS blends, rescaled using equation (8), represent on the corresponding reference curve (v₀ = 11.25 mm s⁻¹). The values of the scaling factors of different extrusion rates can be obtained by the comparison of the extensional ratio at the same extensional stress.⁸ Figure 6 shows the extensional master curve of the blend melts. This manifests from Figure 6 that the extensional master curve proposed from the tests of pure polymer melts⁸ can be also applied to HDPE/PBS blends.

Figure 6.

Extensional master curves of HDPE/PBS blends.

Based on the analysis of the extensional master curve and the original experimental data, at the same extensional stress, the larger the value of the extensional force, the smaller is the scaling factor. On one hand, the macromolecules of polymer melts preorient in the capillary, the extensional force increases,⁸ and the increasing extrusion rate intensifies the macromolecular preoriention, so the extensional force increases accordingly. On the other hand, because of the relatively low external temperature in extensional region, with the extrusion rate increasing, the dwell time shortens in the extensional region for polymer melts, so the cooling degree becomes small and the extensional force decreases accordingly. Thus, the scaling factor decreases with the degree of the macromolecular preorientation increasing, while the scaling factor increases with the increasing cooling degree. Figure 7 shows the dependence of the scaling factor on the extrusion rate. It can be seen from Figure 7 that the macromolecular preorientation of the blends dominates in the test, so the scaling factor reduces with the increasing extrusion rate. Moreover, the relationship between the scaling factor and the extrusion rate obeys the power law, which can be written as:

Figure 7.

Dependence of ξ on v₀.

ξ = a v_{0}^{b}

where a and b are constants.

Predictive ability of extensional viscosity

Approach of extensional master curve

According to the analysis of the extensional master curve, by substituting equation (8) into equations (5) and (7), the predicted extensional strain rate $({\dot{ε}}_{i})$ and the predicted extensional viscosity $(η_{e, i})$ can be written as

{\dot{ε}}_{i} = \frac{v_{0, i} V_{r} e ln (V_{r} ξ)}{L}

η_{e, i} = \frac{η_{e, r} v_{0, r} ln (V_{r})}{ξ v_{0, i} ln (V_{r} ξ)}

The values of a and b in equation (9) are determined from the known extensional curve, and then the scaling factor of the extensional curve at the unknown extrusion rate is obtained from equation (9). So, the scaling factor can be obtain from equation (9) at the extrusion rate of 45 mm s⁻¹, if the extensional curves of HDPE melt at the extrusion rates of 11.25, 22.5, and 33.75 mm s⁻¹ are available.

Approach of back propagation neural network model

As shown in Figure 8, the back propagation neural network model is usually made up of three parts: an input layer, one or more hidden layers, and an output layer. The training data were introduced into the input layer, and then propagated to the hidden layer, and finally to the output layer in the network. In the transfer process, log-sigmoid logsig function and linear purelin function are chosen as transfer functions, training function was trainlm that updates bias and weight values according to Levenberg–Marquardt optimization.²² For HDPE/PBS melts, the two input variables were the extensional strain rate and the extrusion rate, and the output variable is the extensional viscosity, so the neuron number in the input layer was 2. The neuron number in the hidden layer was 3 according to the empirical equation.²² Moreover, the predictions based on the back propagation neural network model are obtained with the help of Matlab software.

Figure 8.

Schematic representation of back propagation neural network architecture.

Approach of constitutive equation

The authors Yang et al.¹⁰ proposed the functions of the creation and destruction of polymer junctions and introduce a movement expression to describe the nonaffine movement of network junctions. Based on network theory, a single-mode constitutive equation is presented as follows:

\begin{array}{l} g τ (1 + {\dot{γ}}_{0}^{a}) + η_{s} τ_{(1)} - \frac{[η_{s}^{2} (t r τ) (b - 6) - 3 b g η_{0} η_{s} (1 + {\dot{γ}}_{0}^{a})] (\dot{γ} \cdot τ + τ \cdot \dot{γ})}{b^{2} g η (1 + {\dot{γ}}_{0}^{a})} \\ = - [\frac{b g η_{0} (1 + {\dot{γ}}_{0}^{a})}{b - 6} + \frac{2 η_{s} (t r τ) (b - 6) - 6 b g η_{0} (1 + {\dot{γ}}_{0}^{a})}{b (b - 6)}] \dot{γ} \end{array}

where η₀ is the zero-shear viscosity, a is the shear-thinning parameter, b is the extensibility parameter for the segments, g is the parameter of the relaxation modulus,

τ_{(1)}

is the upper convected derivative of

τ

and is given by:

τ_{(1)} = (D τ / D t) - k \cdot τ - τ \cdot k^{T}

where D/Dt is the material derivative. The values of η₀, a, b, and g are got by fitting the experimental data of shear viscosity. In this article, the values of η₀, a, b, and g are 15,305 Pa·s, 0.2, 45, and 2700 Pa for HDPE melt, respectively, and the values of

η_{0}

, a, b, and g are 13,580 Pa·s, 0.25, 36, and 2500 Pa for the blend melts with PBS content of 20%, respectively.

Figure 9(a) to (c) shows the comparison between measured extensional viscosity at the extrusion rate of 45 mm s⁻¹ for HDPE melt and the predictions of extensional master curve, back propagation neural network model, and constitutive equation. Figure 10 (a) to (c) shows the comparison between measured extensional viscosity at the extrusion rate of 45 mm s⁻¹ for HDPE/PBS blends $(ϕ = 20 %)$ and the predictions of extensional master curve, back propagation neural network model, and constitutive equation. It can be found from Figures 9 and 10 that the predictions of the back propagation neural network model close to the experiment results, while the predictions based on the extensional master curve analysis and the constitutive equation deviate from the experiment results as the extensional strain rate is smaller than 1 s⁻¹. In general, the back propagation neural network model mimics a lot of the known information to predict the unknown nonlinear parameters based on error back propagation training algorithm. The extensional master curve depicts simply the dependence of the extensional curve on the extrusion rate based on the similarity of extensional curves, Moreover, the predictions based on the single-mode constitutive equation are close to the experimental data for the flow properties of low-density polyethylene melt,¹⁰ but the linear polymer and the branched polymer have different macromolecular structures, the single-mode constitutive equation, suitable for the branched polymer, may be unsuitable for the linear polymer. Thus, the predictions based on the extensional master curve analysis and the constitutive equation deviate from the experiment results for some polymer melts.

Figure 9.

Comparison between measured extensional viscosity at extrusion rate of 45 mm s⁻¹ for the HDPE melt and the predictions of (a) extensional master curve, (b) back propagation neural network, and (c) constitutive equation.

Figure 10.

Comparison between measured extensional viscosity at extrusion rate of 45 mm s⁻¹ for the HDPE/PBS blends $(ϕ = 2 0 %)$ and the predictions of (a) extensional master curve, (b) back propagation neural network, and (c) constitutive equation.

Without simplified assumptions, using this approach of the back propagation neural network model makes it possible to obtain online the approximate experimental data of high extrusion rates and wide extensional strain rates for the purposes of polymer processing control and monitoring.

Conclusions

The effects of the processing conditions and the mixing ratios on the flow behavior of HDPE/PBS blends were significant under the experimental conditions. The melt shear flow of the blends obeyed the power law. The influence of the mixing ratios on the non-Newtonian properties was obvious. PBS melt exhibited almost Newtonian behavior, and the non-Newtonian behavior of the blends weakened with the addition of PBS melt. The dependence of the shear viscosity on temperature was consistent with Arrhenius equation. The melt shear viscosity of the blends decreased sharply with the increasing shear rate. The melt shear viscosity decreased nonlinearly with increasing PBS content, and the variation of the shear viscosity with PBS content and temperature at low apparent shear rates was more significant than that at high apparent shear rates. The extensional viscosity of the blends decreased with the increasing extensional strain rate, the extrusion rate, and PBS content. The extensional flow curves at different extrusion rates overlapped to the reference curve using the scaling factor, and the scaling factor decreased with the extrusion rate increasing according to the power law. Compared with the predictions of the extensional master curve and the constitutive equation, the extensional viscosity predictions of the back propagation neural network model closed to the experimental results. Thus, it is convenient to obtain the extensional viscosity for the purposes of polymer processing control and monitoring using this approach of the back propagation neural network model.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Science and Technology Key Research Project of Education Department of Henan Province (No. 14B430013 and No. 14A150016) and the Doctoral Fund Project of Henan Polytechnic University (No. B2013-016).

References

Ray

Okamoto

. Structure and properties of nanocomposites based on poly (butylene succinate) and organically modified montmorillonite, J Appl Polym Sci 2006; 102: 777–785.

Bhatia

Gupta

Bhattacharya

An investigation of melt rheology and thermal stability of poly (lactic acid)/poly (butylene succinate) nanocomposites, J Appl Polym Sci 2009; 114: 2837–2847.

Wang

Zhang

Toughening modification of PLLA/PBS blends via in situ compatibilization, Polym Eng Sci 2009; 49: 26–33.

Liang

Zhong

. Estimation of elongation viscosity of polyethylene melts. J Thermoplast Compos Mater 2011; 24: 367–376.

Uematsu

Tanoue

Lemoto

Development of polymer extrusion extruder for evaluating with a small amount of polymeric material. Int Polym Process 2012; 27: 434–441.

Wagner

Collignon

Verbeke

. Rheotens-mastercurves and elongational viscosity of polymer melts. Rheol Acta 1996; 35: 117–126.

Muke

Ivanov

Kao

Extensional rheology of polypropylene melts from the Rheotens test. J Non-Newton Fluid Mech 2001; 101: 77–93.

Wagner

Bernnat

Schulze

. The rheology of the rheotens test. J Rheol 1998; 42: 917–928.

Liang

Zhong

. Prediction of elongation viscosity of polymer melts. Polym Eng Sci 2010; 50: 2190–2193.

10.

Yang

Liang

Wang

. A single-mode rheological model for steady flows of polymer melts. J Appl Polym Sci 2012; 125: 3107–3114.

11.

Bridgwater

. Prediction of extrusion pressure using an artificial neural network. Powder Technol 2000; 108: 65–73.

12.

Petrova

Kazmer

. Incorporation of phenomenological models in a hybrid neural network for quality control of injection molding. Polym-Plast Technol Eng 1999; 38: 1–18.

13.

Zweiri

Seneviratne

Althoefer

. Stability analysis of a three-term backpropagation algorithm. Neural Netw 2005; 18: 1341–1347.

14.

Feng

Shen

Preparation and properties of PBS/sisal-fiber composites. Polym Eng Sci 2011; 51: 474–481.

15.

Zhao

Feng

Mechanical and thermal properties of epoxidized soybean oil plasticized polybutylene succinate blends. Polym Adv Technol 2012; 23: 632–636.

16.

Reddy

Nama

Yang

. Polylactic acid/polypropylene polyblend fibers for better resistance to degradation. Polym Degrad Stab 2008; 93: 233–241.

17.

Hamad

Kaseem

Deri

. Rheological and mechanical properties of poly (lactic acid)/polystyrene polymer blend. Polym Bull 2010; 65: 509–519.

18.

Hamad

Kaseem

Deri

. Rheological and mechanical characterization of poly (lactic acid)/polypropylene polymer blends. J Polym Res 2011; 18: 1799–1806.

19.

Laun

Schuch

. Transient elongational viscosities and drawability of polymer melts. J Rheol 1989; 33: 119–175.

20.

Wang

. Technology of polymer modification. Beijing, China: Chemical Industry Press, 2004.

21.

Wagner

Schulze

Göttfert

. Rheotens-mastercurves and drawability of polymer melts. Polym Eng Sci 1996; 36: 925–935.

22.

Han

. Artificial neural network theory, design and application. Beijing, China: Chemical Industry Press, 2002.

Flow properties and extensional viscosity prediction of high-density polyethylene and poly(butylene succinate) blends

Abstract

Keywords

Introduction

Experimental

Raw materials

Processing of blends

Instrument and methodology

Results and discussion

Shear flow property

Dependence of melt shear viscosity on shear rate and temperature

Dependence of melt shear viscosity on temperature and mass fraction of PBS

Extensional flow property

Dependence of extensional viscosity on extensional strain rate

Dependence of extensional viscosity on extrusion rate and PBS content

Analysis of extensional master curve

Predictive ability of extensional viscosity

Approach of extensional master curve

Approach of back propagation neural network model

Approach of constitutive equation

Conclusions

Footnotes

Declaration of Conflicting Interests

Funding

References