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Lee, Mao Ye and Wang-Chien Lee",authors:[{id:"116026",title:"Dr.",name:"Ken",middleName:null,surname:"Lee",fullName:"Ken Lee",slug:"ken-lee"},{id:"116027",title:"Mr.",name:"Mao",middleName:null,surname:"Ye",fullName:"Mao Ye",slug:"mao-ye"},{id:"116028",title:"Dr.",name:"Wang-Chien",middleName:null,surname:"Lee",fullName:"Wang-Chien Lee",slug:"wang-chien-lee"}]},{id:"37518",title:"Three-Dimensional Lineament Visualization Using Fuzzy B-Spline Algorithm from Multispectral Satellite Data",slug:"three-dimensional-lineament-visualization-using-fuzzy-b-spline-algorithm-from-multispectral-sate",signatures:"Maged Marghany",authors:[{id:"96666",title:"Prof.",name:"Maged",middleName:null,surname:"Marghany",fullName:"Maged Marghany",slug:"maged-marghany"}]},{id:"37519",title:"COMS, the New Eyes in the Sky for Geostationary Remote Sensing",slug:"coms-the-new-eyes-in-the-sky-for-geostationary-remote-sensing",signatures:"Han-Dol Kim, Gm-Sil Kang, Do-Kyung Lee, Kyoung-Wook Jin, Seok-Bae Seo, Hyun-Jong Oh, Joo-Hyung Ryu, Herve Lambert, Ivan Laine, Philippe Meyer, Pierre Coste and Jean-Louis Duquesne",authors:[{id:"114809",title:"Dr.",name:"Han-Dol",middleName:null,surname:"Kim",fullName:"Han-Dol Kim",slug:"han-dol-kim"},{id:"116949",title:"Dr.",name:"Gm-Sil",middleName:null,surname:"Kang",fullName:"Gm-Sil Kang",slug:"gm-sil-kang"},{id:"116950",title:"Mr.",name:"Pierre",middleName:null,surname:"Coste",fullName:"Pierre Coste",slug:"pierre-coste"},{id:"150218",title:"Mr.",name:"Do-Kyung",middleName:null,surname:"Lee",fullName:"Do-Kyung Lee",slug:"do-kyung-lee"},{id:"150219",title:"Dr.",name:"Kyoung-Wook",middleName:null,surname:"Jin",fullName:"Kyoung-Wook Jin",slug:"kyoung-wook-jin"},{id:"150221",title:"Mr.",name:"Seok-Bae",middleName:null,surname:"Seo",fullName:"Seok-Bae Seo",slug:"seok-bae-seo"},{id:"150222",title:"Mr.",name:"Herve",middleName:null,surname:"Lambert",fullName:"Herve Lambert",slug:"herve-lambert"},{id:"150223",title:"Mr.",name:"Ivan",middleName:null,surname:"Laine",fullName:"Ivan Laine",slug:"ivan-laine"},{id:"150224",title:"Mr.",name:"Philippe",middleName:null,surname:"Meyer",fullName:"Philippe Meyer",slug:"philippe-meyer"},{id:"150225",title:"Mr.",name:"Jean-Louis",middleName:null,surname:"Duquesne",fullName:"Jean-Louis Duquesne",slug:"jean-louis-duquesne"},{id:"150227",title:"Mr.",name:"Hyun-Jong",middleName:null,surname:"Oh",fullName:"Hyun-Jong Oh",slug:"hyun-jong-oh"},{id:"150228",title:"Dr.",name:"Joo-Hyung",middleName:null,surname:"Ryu",fullName:"Joo-Hyung Ryu",slug:"joo-hyung-ryu"}]},{id:"37520",title:"Hyperspectral Remote Sensing - Using Low Flying Aircraft and Small Vessels in Coastal Littoral Areas",slug:"hyperspectral-remote-sensing-using-low-flying-aircraft-and-small-vessels-in-coastal-littoral-are",signatures:"Charles R. Bostater, Jr., Gaelle Coppin and Florian Levaux",authors:[{id:"115690",title:"Dr.",name:"Charles",middleName:null,surname:"Bostater",fullName:"Charles Bostater",slug:"charles-bostater"}]},{id:"37521",title:"CSIR - NLC Mobile LIDAR for Atmospheric Remote Sensing",slug:"csir-nlc-south-africa-mobile-lidar-for-atmosphere-remote-sensing",signatures:"Sivakumar Venkataraman",authors:[{id:"103080",title:"Prof.",name:"Sivakumar",middleName:null,surname:"Venkatarman",fullName:"Sivakumar Venkatarman",slug:"sivakumar-venkatarman"}]},{id:"37522",title:"Active Remote Sensing: Lidar SNR Improvements",slug:"active-remote-sensing-lidar-snr-improvements",signatures:"Yasser Hassebo",authors:[{id:"111431",title:"Dr.",name:"Yasser",middleName:null,surname:"Hassebo",fullName:"Yasser Hassebo",slug:"yasser-hassebo"}]},{id:"37523",title:"Smart Station for Data Reception of the Earth Remote Sensing",slug:"smart-station-for-data-reception-of-the-earth-remote-sensing",signatures:"Mykhaylo Palamar",authors:[{id:"108533",title:"Dr.",name:"Mykhaylo",middleName:"Ivanovich",surname:"Palamar",fullName:"Mykhaylo Palamar",slug:"mykhaylo-palamar"}]},{id:"37524",title:"Atmospheric Propagation of Terahertz Radiation",slug:"atmospheric-propagation-of-terahertz-radiation",signatures:"Jianquan Yao, Ran Wang, Haixia Cui and Jingli Wang",authors:[{id:"106272",title:"Dr.",name:"Ran",middleName:null,surname:"Wang",fullName:"Ran Wang",slug:"ran-wang"},{id:"112459",title:"Prof.",name:"Jianquan",middleName:null,surname:"Yao",fullName:"Jianquan Yao",slug:"jianquan-yao"},{id:"112460",title:"Dr.",name:"Haixia",middleName:null,surname:"Cui",fullName:"Haixia Cui",slug:"haixia-cui"},{id:"112462",title:"Dr.",name:"Jingli",middleName:null,surname:"Wang",fullName:"Jingli Wang",slug:"jingli-wang"}]},{id:"37525",title:"Road Feature Extraction from High Resolution Aerial Images Upon Rural Regions Based on Multi-Resolution Image Analysis and Gabor Filters",slug:"road-feature-extraction-from-high-resolution-aerial-images-based-on-image-classification-and-gab",signatures:"Hang Jin, Marc Miska, Edward Chung, Maoxun Li and Yanming Feng",authors:[{id:"115888",title:"Dr.",name:"Hang",middleName:null,surname:"Jin",fullName:"Hang Jin",slug:"hang-jin"},{id:"137590",title:"Dr.",name:"Marc",middleName:null,surname:"Miska",fullName:"Marc Miska",slug:"marc-miska"},{id:"137591",title:"Prof.",name:"Edward",middleName:null,surname:"Chung",fullName:"Edward Chung",slug:"edward-chung"},{id:"137592",title:"Dr.",name:"Maoxun",middleName:null,surname:"Li",fullName:"Maoxun Li",slug:"maoxun-li"},{id:"137593",title:"Prof.",name:"Yanming",middleName:null,surname:"Feng",fullName:"Yanming Feng",slug:"yanming-feng"}]},{id:"37526",title:"Hardware Implementation of a Real-Time Image Data Compression for Satellite Remote Sensing",slug:"hardware-implementation-of-a-real-time-image-data-compression-for-satellite-remote-sensing",signatures:"Albert Lin",authors:[{id:"106741",title:"Mr.",name:"Albert",middleName:null,surname:"Lin",fullName:"Albert Lin",slug:"albert-lin"}]},{id:"37527",title:"Progress Research on Wireless Communication Systems for Underground Mine Sensors",slug:"progress-research-on-wireless-communication-systems-for-underground-mine-sensors",signatures:"Larbi Talbi, Ismail Ben Mabrouk and Mourad Nedil",authors:[{id:"113351",title:"Prof.",name:"Larbi",middleName:null,surname:"Talbi",fullName:"Larbi Talbi",slug:"larbi-talbi"}]},{id:"37528",title:"Cold Gas Propulsion System - An Ideal Choice for Remote Sensing Small Satellites",slug:"cold-gas-propulsion-system-an-ideal-choice-for-remote-sensing-small-satellites",signatures:"Assad Anis",authors:[{id:"111486",title:"Prof.",name:"Assad",middleName:null,surname:"Anis",fullName:"Assad Anis",slug:"assad-anis"}]}]}]},onlineFirst:{chapter:{type:"chapter",id:"62131",title:"Mathematical Modeling and Numerical Optimization of Composite Structures",doi:"10.5772/intechopen.78259",slug:"mathematical-modeling-and-numerical-optimization-of-composite-structures",body:'Carbon fiber reinforced plastics (CFRP) are the most promising modern composite materials. High-duty structures used in aviation and space industry, car manufacturing and building sector require new CFRPs as well as ways to improve their characteristics. Applying computer modeling techniques significantly reduces both the time and cost of investigations aimed at searching optimal parameters of CFRP structures [1]. Mathematical modeling provides an opportunity for comprehensive analysis of both CFRPs and CFRP structures. It has become an effective tool for solving important applied problems.
To build a mathematical model of composite materials, including those made of carbon fibers, one relies upon the experimental data acquired in mechanical testing. Wide application of digital testing machines has brought such experiments to higher level of quality. By measuring a large number of parameters with a high discretization frequency, modern testing machines allow for high amount of information on material deformation and failure to be obtained within a single experiment. Therefore, data processing has become an important step for mathematical modeling of CFRPs and CFRP structures. This is exceptionally important because of quite specific behavior of CFRPs and of their components: fibers and matrices.
One of the features of such materials is their different strength and stiffness behavior in tension and compression combined with nonlinearities of stress-strain curves. Multiple studies for epoxy matrices showed that their ultimate strains in tension were much lower than in compression: approximately 4 versus 20% and more [2]. Moreover, under tension and compression, the deformation behavior of epoxy matrices significantly differs. The corresponding stress-strain curves have different stiffness (secant modulus) at the same values of strain. The similar difference can be observed for CFRPs. In [3, 4, 5], it was shown that in tension tests of carbon fiber specimens with reinforcement angles less than
Contrasting behavior in tension and compression, stiffening, softening and other nonlinearities are forcing researchers to build and use special mathematical models and computing algorithms. Mathematical models taking into account the abovementioned properties of materials were proposed and studied theoretically by Timoshenko [6] and Ambartsumyan [7, 8] in the mid-twentieth century. Later, Jones had experimentally, theoretically and numerically studied the nonlinear behavior of several fiber-reinforced composites. The main focus of the research was on the difference in stiffness and strength behavior under tension and compression [9]. After Ambartsumyan’s and Jones’ researches, a lot of studies were dedicated to this problem. Most of them were dealing with linear bi-modulus models of materials or 3D finite elements. In [10], Ambartsumyan with a coauthor suggested a theoretical approach to modeling of multimodulus nonlinear elastic beams under bending, but still without calculations.
Another trend is studying the behavior of sandwich panels or beams with a CFRP faces having differences in tension and compression along with the other mentioned nonlinearities [11, 12, 13]. These works concern the problem of flexure of CFRPs and similar materials. They consider bending of specimens as a reference test. The first paper [11] is devoted to experimental investigations and shows most of the nonlinearities we supposed such materials should have: stiffening in tension, softening in compression, different moduli even at the origin of coordinates. Other two works [12, 13] present more complex studies including full cycle of mathematical modeling spanning from the experimental investigations to numerical ones.
A comprehensive approach to modeling and simulation of nonlinear elastic deformation of polymer matrices and different CFRPs was presented in [14]. This chapter deals with different strength and stiffness behavior of the materials in tension and compression exemplified by a case of three-point bending. This approach implements a full cycle of model development and validation, which comprises the following stages: carrying out tests and acquiring experimental data, data prepossessing and building stress-strain curves, analytical approximation of acquired curves, mathematical modeling and numerical simulation of deformation processes, comparative analysis of results of numerical modeling to acquired data.
For most of the composite materials models, we can write the relations between average stresses
where
The structural model of fiber reinforced composite described in [15, 16, 17, 18] has become a foundation for a large number of current researches. Now it is widely used while simulating the behavior of composite structures. The model is based on the following assumptions: the stress-strain state into isotropic elastic fibers and into entire volume of isotropic ideally elastic matrix is homogeneous; fibers and matrix are deformed jointly along the direction of reinforcement; stresses in fibers and in matrix corresponding to other directions are equal.
For computing the effective elastic modulus of unidirectional fiber-reinforced composite, the Reuss-Voigt average was used giving the following formulae
where all the terms having squared Poisson coefficients are neglected.
Herewith
On the ground of symmetry of compliance tensor, one has
Formulae for effective coefficients of thermal expansion have the following form
In description of the model, it is noted that among formulae for effective moduli, those obtained using Reuss averaging (in particular formulae for
gives more accurate approximation than (Eq. (2)) does. Hereinafter share moduli of matrix and fibers are
Components of effective stiffness tensor for unidirectionally reinforced layer in case of state of plane stress have the following form:
Unwritten expressions can be obtained using symmetry rule or vanish. Hereinafter, we assume
The coefficients in the relations (Eq. (1)) for example are defined by the formulas given in [1, 17, 18].
Three-point bending flexural test has been one of the standard techniques to determine physical and mechanical characteristics of materials. Figure 1 shows a scheme of physical model of three-point bending of a beam with the rectangular cross section
Three-point bending of a rectangular-sectioned beam.
In this case, the beam’s upper part undergoes compression strain in the longitudinal direction, bottom part—tension strain. VSE-1212 polymer matrix and VKU-28 (T-800 carbon yarn plus VSE-1212 epoxy matrix) structured CFRP react differently to tension and compression. VKU-28 has been one of the most promising types of CFRPs that is going to be used in the latest generations of aircrafts. The effect of accounting for this factor on the computational results is essential. Further, these results are compared to acquired ones.
Due to very low deformation rates, the classical theory of beam bending can be regarded as satisfactory for description of the equilibrium state. To this end, it is convenient to consider the beam’s median surface as a reference one.
The beam’s stress-strain state is characterized by the following values determined on the reference surface: the shear force
The reactions
Strain distribution for the beam’s thickness can be obtained from the Kirchhoff-Love kinematic hypotheses:
where
The constitutive equation can be expressed as:
where the superscript “+” refers to the areas with positive strains and “–” – to the area with negative ones;
The longitudinal force
Having substituted (Eq. (12)) with the relations (Eq. (8)), (Eq. (10)), (Eq. (11)) into and integrated it over the beam thickness, one obtains a system of equations to determine
at
at
The system of equations (Eq. (13)) and (Eq. (14)) in general case is nonlinear, but in the case of piecewise linear constitutive equations which take into account different strength and stiffness behavior in tension and compression expressed as follows:
it can be solved analytically. In the nonlinear case, the Newton method is applied to solve the equations (Eq. (13)) and (Eq. (14)), and then, the linearized system
can be solved for unknown values
where
As the initial approximation at small values of the load
Having determined the change of median surface curvature from the equations (Eq. (13)) and (Eq. (14))
one can write down a differential equation to determine the beam bend. For that purpose, the bend function is expressed as follows:
Using the equation (Eq. (9)) and the beam’s fixing conditions, a system of equations can be derived:
The solution of these equations can be obtained using the methods of solving boundary-value problems for systems of ordinary differential equations. For that purpose, the modified collocation and least-residuals method [19, 20, 21] were applied.
Numerical analysis of deformation processes in VSE-1212 polymer matrix and VKU-28 structured carbon fiber is based on approximation of stress-strain curves and the three-point bending model.
Further, three different specimens with the geometrical sizes
specimen 1—VSE-1212 polymer matrix,
specimen 2—VKU-28 structured carbon fiber (the specimen was cut out along the reinforcement),
specimen 3—VKU-28 structured carbon fiber (the specimen was cut out perpendicular to the reinforcement),
In Figure 2, one can see the simulation results for beam three-point bending, obtained through different approaches to approximation of the constitutive equations, and their comparison with the experimental data.
Experimental (solid curves) and dependencies of beam-deflection and load obtained in simulation: linear approximation (1); quadratic approximation by a polynomial of the second degree (2); cubic approximation (3); linear and power-law approximation (4); a–c are specimens 1 − 3 respectively; d—the solution to a three-point bending problem without account for the different strength and stiffness behavior in tension (curve 1) and compression (curve 2). The solid line shows the results of mechanical tests.
Applying the linear dependencies to tension and compression has not resulted in adequate approximation even for 30% of curve. Using more complex than quadratic approximation laws at first led to a significant deviation from the experimental curve and then to divergence of the Newton method iteration process. This is explained by the fact that in tension tests, due to specimens’ fragility, the strain range for the polymer matrix specimens was limited to 2%, while in the bending tests, the strains in tension zone reached 4–5%.
Thus, to solve the bending problem, the tension curve was extrapolated into the domain of high strains. The extrapolations obtained using a polynomial of the third degree, and by linear and power-law function reached the maximum too quickly and then started to decrease, which is against the physics behind the deformation process. A similar effect was observed when calculating the bending of the carbon-fiber specimens cut out along direction of reinforcement filler.
The calculations using quadratic approximation and extrapolation of tension curves and approximation of compression curves within a short (up to 6%) segment have turned out to be best for qualitative and quantitative description of the nonlinear character of VSE-1212 polymer matrix bending. In the case of the specimen cut out perpendicular to direction of its reinforcement, all the approximations have shown the results close to experiment. At the same time for the test with the maximum load, the best option has still been application of quadratic approximations.
Taking different strength and stiffness behavior in tension and compression into account has an essential effect. As it was demonstrated earlier, the tension tests of VKU-28 specimens produced nonlinear stress-strain curves, while the difference of characteristics between tension and compression reached 5–7% for the longitudinal reinforcements and 12–15%—for the transverse ones (see Tables 1 and 2).
Approximation type | Approximation coefficients | MSD | ||
---|---|---|---|---|
Tension of VKU-28 CFRP, | ||||
A1 | 160.8 | 1.4e – 2 | ||
A2 | 144.9 | 1.44e + 3 | 5.7e – 4 | |
A3 | 144.0 | 1.66e + 3 | –1.14e + 13 | 4.3e – 4 |
A4 | 143.0 | 8.87e + 2 | 1.87 | 4.1e – 4 |
Compression of VKU-28 CFRP, | ||||
A1 | 155.4 | 5.8e – 3 | ||
A2 | 160.2 | –3.33e + 3 | 3.4e – 3 | |
A3 | 155.9 | 4.31e + 3 | −3.00e + 15 | 2.9e – 3 |
A4 | 157.7 | −5.81e + 8 | 3.98 | 3.0e – 3 |
Approximation coefficients for stress-strain curves of VKU-28 carbon fiber specimens reinforced in longitudinal direction and mean square deviation (MSD) of
Approximation type | Approximation coefficients | MSD | ||
---|---|---|---|---|
Tension of VKU-28 CFRP, | ||||
A1 | 7.37 | 1.1e – 2 | ||
A2 | 7.89 | −9.22e + 1 | 6.4e – 4 | |
A3 | 7.87 | −8.97e + 1 | 2.54e + 11 | 4.3e – 4 |
A4 | 7.82 | −2.23e + 2 | 2.20 | 4.8e – 4 |
Compression of VKU-28 CFRP, | ||||
A1 | 8.90 | 6.7e – 3 | ||
A2 | 9.21 | −1.20e + 2 | 4.1e – 3 | |
A3 | 8.96 | 1.16e + 2 | −5.09e + 13 | 3.7e – 3 |
A4 | 9.04 | −5.16e + 6 | 3.95 | 3.7e – 3 |
Approximation coefficients for stress-strain curves of carbon fiber specimens reinforced in transverse direction and mean square deviation (MSD) of
For the polymer matrix this difference exceeded 15% (see Tables 3 and 4).
Approximation type | Approximation coefficients | MSD | ||
---|---|---|---|---|
Strain of VSE-1212 polymer matrix (Constant cross section) | ||||
A1 | 3.30 | — | — | 2.9e − 2 |
A2 | 3.90 | −4.38e + 1 | — | 1.5e − 3 |
A3 | 3.83 | −3.17e + 1 | −4.94e + 11 | 6.7e − 4 |
A4 | 3.80 | −1.05e + 2 | 2.25 | 6.6e − 4 |
Strain of VSE-1212 polymer matrix (Variable cross section) | ||||
A1 | 3.33 | 2.7e − 2 | ||
A2 | 3.89 | −4.02e + 1 | 1.8e − 3 | |
A3 | 3.80 | −2.48e + 1 | −6.30e + 2 | 4.0e − 4 |
A4 | 3.77 | −1.40e + 2 | 2.35 | 2.7e − 4 |
Approximation coefficients for tension curves of VSE-1212 polymer matrix,
Approximation type | Approximation coefficients | MSD | ||
---|---|---|---|---|
Compression of VSE-1212 polymer matrix (Constant cross section), | ||||
A1 | 0.77 | 2.3e − 1 | ||
A2 | 1.60 | −3.97 | 8.2e − 2 | |
A3 | 2.36 | −1.29e + 1 | 2.37e + 10 | 1.4e − 2 |
A4 | −5.71 | 5.49 | 0.90 | 3.8e − 2 |
Compression of VSE-1212 polymer matrix (Variable cross section), | ||||
A1 | 0.69 | 3.5e − 1 | ||
A2 | 1.69 | −5.22 | 1.4e − 1 | |
A3 | 2.71 | −1.84e + 1 | 3.84e + 1 | 3.6e − 2 |
A4 | −2.07 | 1.72 | 0.72 | 4.8e − 2 |
Compression of VSE-1212 polymer matrix | ||||
A1 | 2.10 | 7.2e − 2 | ||
A2 | 3.05 | −2.12e + 1 | 4.2e − 3 | |
A3 | 3.18 | −2.85e + 1 | 9.13e + 1 | 1.1e − 3 |
A4 | 3.31 | −1.24e + 1 | 1.75 | 1.9e − 3 |
Approximation coefficients for compression curves of VSE-1212 polymer matrix and mean square deviation (MSD) of
Tables 1–4 show approximation results for the above-presented stress-strain curves by different functions at different intervals:
by the linear approximation
by the polynomial of the second degree
by the polynomial of the third degree
by a combination of linear and power-law functions
However, if bending tests have been performed to determine an elasticity module of CFRPs, different strength and stiffness behavior in tension and compression is compensated and one obtains some averaged characteristic.
It is useful to consider the effect of the way for determining and setting of the mechanical characteristics on modeling of three-point bending of the carbon-fiber beam cutout perpendicular to its reinforcements. Figure 2d shows the solutions obtained while using a linear approximation of the constitutive equations with equal elastic moduli for tension and compression: for curve 1 the modulus was obtained from tension experiments, for curve 2—from compression ones (see Table 2).
As one can see the calculated linear results without account for the different strength and stiffness behavior in tension and compression have differed from the results of mechanical tests (the solid curve) by more than 15%.
Most of the real CFRP structures under day-to-day service conditions bear complex loads that result in formation of tension, compression and bending zones as well as their combinations in the structures. Applying the traditional methods for determination of material characteristics in combination with linear deformation models (in particular those that do not account for the different strength and stiffness behavior in tension and compression) for calculation of such structures, one risks to distort the deformation and stress pattern significantly, which, in its turn, results in either underestimation or overestimation of the structure’s strength and rigidity. Keeping in mind that carbon fibers are used for manufacturing of high-duty structures, their computation demands different strength and stiffness behavior in tension and compression to be taken into account.
Composite overwrapped pressure vessels (COPV) are used in the rocket and spacecraft making industry due to their high strength and lightweight. Consisting of a thin, nonstructural liner wrapped with a structural fiber composite COPV are produced to hold the inner pressure of tens and hundreds atmospheres. COPV have been one of the most actual and perspective directions of research, supported especially by NASA [22, 23].
Designing of a highly reliable and efficient COPV requires a technology for analysis of its deformation behavior and strength assessment. This technology should allow one to obtain target COPV parameters through changing vessel’s geometry, structural and mechanical material parameters while keeping its useful load.
Application of combination mathematical modeling and numerical optimization makes it possible to reduce the cost and the duration of identifying the best parameters for a COPV. However, this approach is characterized by a number of hurdles. Overcoming these hurdles determines the success of an optimum designing of such structures.
So far, there have been two main approaches in optimization of composite structures: analytical and numerical ones.
In the first approach, the problems are solved basing on their simplified statement, for example using the momentless (membrane) shell theory and the netting model of composite material (CM) [24, 25, 26, 27]. The obtained results may be far from reality; however, they are of value for testing of numerical optimization methods.
Application of the numerical approach in designing, on the other hand, produces a number of challenges that must be overcome, for example, lack of reliable methods for global optimization; nonconvexity and nonlinearity of constraint functions; ill-conditioned boundary value problems; different scaling of optimization criteria represents just some of the obstacles that prevent from reliable optimization of COPV.
Numerical analysis is usually a computation-intensive process and takes considerable time. One way to solve this problem is approximation of the objective function using different approaches, such as response surface method [28] and neural network [29]. Some kinds of numerical analyses use a small number of design variables, functions and/or corresponding set of their discrete values (analytical geometry parametrization [30], finite set of feasible winding angles [31]).
Another way is reasonable simplification of the elasticity problem statement, for example by using the membrane theory or other shell theories [30, 32, 33], that leaves the question of results validity. This is the approach we have applied in our study. For validation, we have used the Timoshenko [34] and Andreev-Nemirovskii [35] shell theories, accounting transverse shears with different degrees of accuracy.
Of course, it should be taken into account that the computed solutions are not optimum in the strict mathematical sense. However, these solutions could provide the considerable economy of the weight while keeping the required strength, and, therefore, they have high engineering value.
Let us consider a multilayer composite pressure vessel at a state of equilibrium under equidistributed inner pressure. We need to determine the parameters of structure and CM meeting the following requirements:
where
We define the optimization problems the following way: to find extremum of one functional from (Eq. (17)) under other constraints.
The structural optimization problem statement includes selection of objective functional, formulation of constitutive equations and constraints on performance and design variables.
The mathematical models describing the vessel’s state are based on the following assumptions:
the vessel is a multilayer thin-walled structure;
the vessel’s layers can have different mechanical characteristics;
the reinforced layer’s material is quasi-homogeneous;
the vessel’s main loading is high inner pressure.
These assumptions allow us to reduce dimension of the corresponding mathematical problem and to build the mathematical vessel’s models based on the different theories of multilayer nonisotropic shells.
Let us consider the vessel as a shell rigidly compressed on the edge. Taking into account a symmetry plane in the middle of the vessel, it is enough to calculate and design only its one half. The type of loading and boundary conditions allows considering the axisymmetric problem statement.
The shell is set by rotation of the generatrix
Shell of rotation geometry.
The Kirchhoff-Love shell theory [36] (KLST) and the improved Timoshenko [34] (TiST) and Andreev-Nemirovskii [35] (ANST) theories are used to solve the direct calculation problems of multilayer composite vessels, to analyze their behavior and to verify optimization problem solutions. The full systems of equations were described in the paper [17].
Relations between stresses and strains are described by the structural models [18]. The main idea of these models is that CM parameters are calculated through matrix and fibers mechanical parameters, fibers volume content and winding angles. The stress-strain state of matrix and fibers is evaluated through stresses and strains of the composite shell. A failure criterion is applied for every component of CM. Here we use the Mises criterion to determine the first stage of failure.
The objective function whose minimum is required is the minimum mass:
where
We chose the following design functions: the curvature radius
The solution has to satisfy the constraints on the shell’s inner volume:
and the strength requirement:
where
We used the following constraints on the design functions:
The method of the continuous geodesic winding has been widely used in the manufacturing of composite shells of revolutions. In this case the winding angles are defined by the Clairaut’s formula:
where
which has the singularity at the edge where the winding angle has to be equal to
We did not consider the problem of fibers slippage. The main goal of the study was to demonstrate the potentials of using CM.
Estimation of composite vessel stress-strain state using offered models leads to the solution of boundary value problems for rigid systems of differential equations. These problems are ill-conditioned, and their solutions have pronounced character of thin boundary layers. Numerical analysis was performed by the spline collocation and discrete orthogonalization methods, realized in the COLSYS [37] and GMDO [38] software. These computing tools have proved to be effective in numerical solving of wide range of problems of composite shell mechanics [1].
We investigated the vessel’s deformations by computing its stress-strain state based on the different shell theories. The vessel’s shape was a part of a toroid:
Figure 4 shows the stress-strain state characteristics of the vessel with the thickness
The stress-strain state characteristics of the composite vessel computed using different shell theories. Longitudinal displacement u—dashed curves; deflection w—solid curves. The curves without symbols correspond to KLST simulations, the curves marked with Δ—to those using TiST, and □—to ANST.
It is easy to see that the basic kinematic characteristics coincide both qualitatively and quantitatively. Small differences are observed only for the stresses and deformations near the compressed edge. The maximum results and qualitative difference were obtained for ANST. This is due to accounting for the transverse shears by nonlinear distribution in a thickness of a shell. Earlier it was shown [1] that ANST’s-based results were the closest to the ones of 3D elastic theory in most cases.
The winding angle’s influence on the COPV performance was investigated using parametric analysis. Dependence of the maximum nVMS in the matrix
The winding angle’s influence on the composite vessel stress-strain state. KLST’s results are drawn without marks, TiST — with symbols Δ, ANST — with □.
The calculated values are very close in the area of their minima (Figure 5 left side). The graphs of kinematic function
The range
It was revealed that the winding angles of minimum stresses values were almost insensitive to the thickness variation. The change of
Additionally, we investigated stress–strain state of the vessel (the thickness
The stress-strain state of the vessel (ψ=±43.2), computed using the three shell theories.
Again the difference is visible only in a very small region near the edge, but now this difference is small enough to be neglected. Moreover, the displacement values of the reference surface, the efforts and the moments completely coincide for all the theories.
All the theories (KLST, TiST, ANST) provided similar estimated characteristics of stress-strain state. This vessel was characterized not only by essential decrease of the maximal nVMS in the matrix and the fibers, but also by their uniform distribution along the generatrix. At the same time, the values of bending moments significantly reduced bringing vessel’s stress-strain state close to momentless.
The performed analysis showed that the optimization problem can be solved using rather simple shell theories (KLST, TiST). These theories are characterized by lower computational complexity of corresponding boundary value problem if compared to ANST. It takes from 10 to 20 times less resources.
One can see that the winding angle as a design parameter gives an opportunity to increase the vessel’s strength significantly. The difference between the “best” and “worst” designs can reach 20–35 times comparing their nVMS in the matrix and fibers. The “worst” designs have the winding angle close to
Inverse problems involve not only numerical methods for fast and reliable solving of direct boundary value problems, but also require numerical optimization methods for finding design parameters.
Here we considered conditional optimization problem, including direct constraints on design functions and trajectory constraints on the solution imposed at the end of the interval. The sequential unconstrained optimization is one of the most widespread approaches to solution of such problems. The main idea of the method is terminal functional convolution and multiple solutions of one-criterion problem using different optimization methods [39]. In our study, the modified Lagrange function was used for the convolution.
Hence we sought for solution of a nonconvex problem of finite-dimensional optimization [40] by discretization of design functions. The methods realized in the OPTCON-A software [41] were used to get the corresponding solution.
The considered design with the continuous geodesic winding has been one of COPV widely used in practice [42, 43].
Important additional design characteristic is its “adaptability in manufacturing.” For example, the 5–10 times difference of thickness along the meridian would become a serious obstacle for vessels manufacturing. Thus, designs of nearly minimum mass possessing good properties and satisfying to the given technological constraints could be of great value than optimum without them.
According to Amelina et al. [44], the design with the geodesic continuous winding has the thickness ratio about 10 and large gradient near the edge.
We verified the solutions of optimization problem by substituting the obtained design parameters into the direct problem. In [44] shown that all three theories yielded close results (Figure 7). The difference is noticeable only for ANST in narrow zones (less than
The stress-strain state characteristics of the vessel with the optimized design functions based on the three shell theories. Longitudinal force T11—dashed curves; bending moment M11—solid curves.
Thus, it is possible to use the simplest shell theory to solve such optimization problem and the estimation of stress-strain state will be close to those obtained using more complex theories.
Mathematical models for nonlinear flexural deformation of CFRPs and polymer matrices with account for their different strength and stiffness behavior in tension and compression have been built. A satisfactory match with the results of mechanical tests has been obtained. The study has proved that the nonlinear properties of polymer matrices and carbon fibers should be taken into account when calculating and designing real structures.
The technology of optimization of COPV has been developed. It makes possible to obtain high pressure vessel designs that not only meet such requirements as minimum mass, preset volume and strength, but also possess a number of additional valuable engineering characteristics including stress-strain state close to momentless and almost equally stressed fibers.
Nonconstant design parameters, such as thickness, winding angles and curvature radius of composite shell give the possibility for additional reduction of COPV mass while keeping its strength. The solutions of the optimization problem have been verified by solving the direct problems with obtained design parameters using the classical and improved shell theories.
The study has demonstrated acceptability and convenience of using simple mathematical models based on Kirchhoff—Love and Timoshenko shell theories for numerical solving optimization problems.
The study was supported by project 18-13-00392 of Russian Science Foundation.
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