Abstract PhD THESIS - Universitatea din Craiova · 2019-07-09 · Abstract . PhD THESIS ....

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UNIVERSITY OF CRAIOVA FACULTY OF MECHANICS

DOCTORAL SCHOOL: Radu Voinea FUNDAMENTAL DOMAIN: Engineering Sciences

DOMAIN: Mechanical Engineering

Abstract

PhD THESIS CONTRIBUTIONS TO THE STUDY OF THE

MECHANICAL BEHAVIOUR OF COMPOSITE MATERIALS WITH AUTOMOTIVE

APPLICATIONS Scientific adviser: Prof. PhD. Nicolae Dumitru

PhD student: Eng. Alexandru Bolcu

CRAIOVA 2018

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CONTENT

Summary (Thesis)

Structure and objectives of phd thesis -(5) Chapter 1. Describing the composite materials. In field research 4(8) 1.1. Introduction 4(8) 1.2. Definitions. Generalities 4(8) 1.3. Fields of use 5(9) 1.4. Properties 5(11) 1.5. Classification. Materials used for the creation of a composite

one 5(11)

1.5.1. Materials for matrix 5(12) 1.5.2. Materials used for reinforcement (fibers) 5(12) 1.6 . Studies for mechanical behaviour of composite materials 6(13) 1.7. Studies regarding the mechanical behaviour of multilayered

composites 6(15)

1.8. Studies regarding eco-composite materials 7(16) 1.9. Using composite materials for cars 7(18) Chapter 2. Research of elastic properties and resistance of composite materials

8(25)

2.1. Introduction 8(25) 2.2. Theoretical considerations 8(27) 2.3. Experimental determinations 9(32) 2.3.1. Used aparatus 9(32) 2.3.2. Studies of ununiformities for composite materials

reinforced with synthetic fibers

10(33) 2.3.3. Experimental determinations on composite materials

based on dammar

12(37) 2.3.4. Study of non-uniformities for composite materials

reinforced with natural fibers

13(44) 2.4. Conclusions 14(47) Chapter 3. Vibration research upon slender composite bars 16(53) 3.1. Introduction 16(53) 3.2. Movemenet equation 16(54) 3.3. Damped free vibrations 18(63) 3.3.1. Case of the simply supported bar 19(64) 3.3.2. Method of variable separation 19(66) 3.4. Experimental determinations 20(68) 3.5. Discussions and conclusions 23(75) Chapter 4. Research on vibrations for multilayer composite bars 24(78) 4.1. Introduction 24(78) 4.2. Theoretical considerations 24(79)

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4.3. Free vibrations 26(82) 4.4. Case of sandwich bar 27(84) 4.5. Experimental determinations 29(90) 4.6. Discussions and conclusions 31(98) Chapter 5. Experimental study, modeling and virtual simulations of car part realized from composite materials

32(101)

5.1. Introduction 32(101) 5.2. Determination of own frequnecies by experimental

measurements 32(101)

5.3. Modeling and virtual simulation of a car part made of composite materials

33(107)

5.3.1. Tridimensional model of the fender 33(107) 5.3.2. Mathematical model with finite elements

of the fender 34(110)

5.3.3. Material properties. Boundary conditions 34(111) 5.3.4. Results 35(114) 5.4. Study of the structural stability of the fender by

fluid-structure analysis

36(121) 5.4.1. Analysis of fluid flow around the fender 36(121) 5.4.2. Stability calculus 37(123) 5.5. Conclusions 39(128) Chapter 6. Capitalization of results, contributions and future research

40(129)

6.1. Capitalization of the researched results 40(129) 6.2. Original contributions 40(130) 6.3. Future research paths 41(132) Bibliography 42(134) Annex 1. -(145) Annex 2. -(153) Annex 3. -(156) Annex 4. -(164) Annex 5. -(179) Annex 6. -(191)

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Ch.1. Describing the composite materials. In field research

1.1. Introduction Economic and social needs, the continous search of using less raw materials and energy,

the unrelenting exploatations of the sourounding enviroment by the humans and also the production rate have led to a fast development of new materials and nontraditional technologies.

The development of these new materials and technologies are creating new paths regarding complex analysis and analitical calculus leading thus the way to unknown technical evolution not known yet by humans. By having a multitude of properties , the composite materials can satisfy the needs of researchers that have been presented previously.

1.2. Definitions. Generalities Composite materials represent the future of engineering, being the most advanced class

of materials invented by humans. R. H. Jones presents us composite materials as „ a fusion of two or several materials at a

macroscopic scale with the intentions of forming a third material that is much more useful”. Composite materials combine the properties of two or several constituent materials thus

having a synergetic effect and using the useful properties and dimishing defects. The constituent materials maintain their identity (at least to a macroscopic level) in the composit material, but their fusion generating a material with different properties and behaviour than of the materials used seperatly.

1.3. Fields of use Being charcaterized by their important assets by comparison to traditional materials and

by having properties easy to figure out, composite materials have a privileged place among other materials in top developing technologies such as: car industry, naval industry, comunications, medical technologies, chemical technologies, aerospace,constructions and so on. A general presentation of fields using composite materials is shown in fig. 1.1.

Fig.1.1. Fiels in which composite materials are used [WWW_14]

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1.4. Properties The main advantages of composite materials in contrast to the traditional ones are reveald

by several propetries such as: - specific mass regarding metals, composite epoxy resin reinforced with light fibers such

as bor, silicium or carbon that have a specific weight under 2 kg/dm3 obtaing thus at low weight a big mechanical resistance;

- high fatigue,shock and abrasion resistance; - very good resistance to atmospsherical agents and no corosion problems; - chemical stability; - good reliability, the failiure of one fiber does not mean that the failiure criterion is

reched due to the fact that the fibers do not allow the cracks to evolve; - high traction resistance; - durability and safety when under loads; - small dilatation coefficient; - low production price for making of finite material in most cases; - good damping for vibrations; - easy to use for complicated geometrical shapes.

1.5. Classification. Materials used for the creation of a composite one Due to the great diversity of composite materials that are available, in the speciality

literature there can be several classifications. A common classification that is general accepted is the one provided by R.M.Jones in

which he divides the composite materials in four groups[JON_1999]: - fiber composite materials; - laminate composite materials; - special composite materials; - a combination of several of the three above mentioned. 1.5.1. Materials for matrix The matrix is made of cheap materials, with inferior properties when regarding to

resistance. The matrix assures the spatial arrangement of the reinforcement, assures the protection of it and the equilibrium between the fibers.

Regarding the nature of the material from which the matrix is made we have: - composite materials with the matrix from pastic materials; - composite materials with metallic matrix; - composite materials with mineral matrix. From the plastic materials used for a matrix we can have thermoset plastic materials

which, during manufacturing suffer a chemical polymineralization reaction and the thermoplastic materials which can have a softening when heated and a hardening when cooled, without any chemical reaction.

From the thermorigid materials we can name the poliestherical,epoxy,fenolic,siliconic resins etc. and from thermoplastic the vinilic,polyamide,acrilonythril-butadein-styren ones etc.

1.5.2. Materials used for reinforcement (fibers) In general the term of fiber is used for a solid material, that is elongated, and for which

the ratio between lenght / diameter is a very big one. Thus we can use the material at its maxium regarding the resistance properties if the fibers are arranged on the direction on which the load is applied.

From a phisical point if view the fibers are: - long fibers which have usual the longitudinal dimension equal to the one of the finite

material; - short fibers that can be from several milimeters to a couple of centimeters; - fibers that are arranged in a tuft pattern

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1.6. Studies for mechanical behaviour of composite materials Composite materials that are reinforced with fibers have a good market place because of

their structural aplications but they have a limited hardness.Hybridization with fibers is a promising strategy for enhacing the properties of composite materials. By combining two or several types of fibers, composite hybrid materials offer the best equilibrium regarding the mechanical properties than the one that are not hybrid. The prediction of mechanical properties for such materials is a chalenge because of the synergetic effects between the fibers. In [SWO_2014] and [WIS_2016] is presented a general description of traction, bending shock, fatigue properties and an in depth analysis of the hybridization effects.

One of the points of interst for researchesrs in the field of mechanics is represented by the vibrating behaviour of composite laminated materials. Studies regarding the vibrational and damping behaviour of laminated composite materials that are reinforced with different types of fibers are presented in [KUL_2018] and [DAO_2017]. In [GUE_2016(1)] is presented the compromise between the elasticity modulus and the ability of damping as a function of composition of the laminated composite fibers that are hybrid and in [ASS_2015] has as its main purpose the investigation of secvential loading effects and of hybridization on the propetries of damping for composite epoxy materials with diagonal weaves of carbon.

Corosion of steel reinforcement in concrete structures has a big cost regarding the global resources for the last decades. Bars reinforced with fiber glass (GFPR) are a good alternative to the reinforced steel bars regarding costs. Studies regarding GFRP are presented in [SHE_2018]. Mechanical properties of GFRP bars subjected to high temperatures have been evaluated in HAJ_2018].

1.7. Studies regarding the mechanical behaviour of multilayered composites

A particular place in the study of composite materials is that of sandwich bars formed from several overlaping layers with constant thickness. Most studies refer to the sandwich bars from three layers, the exterior layers having elastic properties and of superior resistance, while the middle ones have viscous-elastic properties.

In [HUA_2017] is investigated the behavior of static failure and low speed impact of sandwich type composites.

[CHE_2017] studies the response at low speed impact of a new sandwich type panel with aluminium foam layers and a wrap of laminated metallic fibers formed from aluminium sheets and of fiber glass type E. In [QIN_2018] is developed an analytical model used to predict the dynamic response of a sqauare sandwich plate, having fixed supports on the edges, a core of metallic foam that is subjected to a transversal hit at low speed by a heavy mass. A similar model but for sandwich bars is used in [QIN_2017]. Also similar approach is used in [ZHA_2016] where it is investigated the dynamic response of sandwich multistratal bars with a shallow metallic foam core, that are totally fixed and are subjected to hits at low speeds from a mass. The interaction between bending and the elongation induced by big deformations is studied also in [QIN_2009] where is studied the large deformations of sandwich beams with a core of metallic foam under a mobile transversal load. Results have also been adapted by WAN_2017] for dynamic response of sandwich asymmetric plates with a aluminium hexagonal honeycomb core subjected to impulsive shock loading, obtaining thus a series of deformations and deterioration.

In [ABO_2017] is studied a sandwich bar formed from a hammer head and a coating of laminar carbon fiber with a aluminium honeycomb core and of rubber that has been used to improve the behaviour at low speed impact. In WAL_2017] is investigated the usage of expandable cork that is a light material and 100% natural, as a core for composite sandwich plates with faces of carbon fiber, following especially the energy absorbing capacity.

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1.8. Studies regarding eco-composite materials The intrest of ecodurable materials in industrial sector (cars,constructions etc.) has lead to

a growing number of scientific works regarding biocomposite materials. [ZUC_2018] proposes a fabrication process that permits the obtaining of unidirectional biocomposites of high quality with a fraction of fiber volume of only 70%.

[KHO_2017] is evaluating the development of biocomposite specific regarding the mechanical properties.

The properties of elongation,bending and impact for fiber biocomposites made from polilactic acid and nettle fibers, made at high pressure have been analized in [KUM_2017]. In [OQL_2014] is presented a complete analysis of polymeric composite materials reinforced with natural fibers.

In [YUS_2016] hybrid composite green materials have used unidirectional arranged fibers made of kenaf, bamboo, coconut to reinforce the polymeric matrix of the polilactic acid. In [VIK_2015] the traction properties, bending, impact resistance have been studied for hybrid composite materials such as synthetic resins reinforced with sisal fibers and ananas fibers unidirectional layered. [SAT_2016] presents the results of experimental investigation of mechanical properties for composite hybrid polyester reinforced with cotton and sisal fibers.

1.9. Using composite materials for cars While delux cars and sports cars have a experience with composite materials for a long

time, there is a trend now for medium budget cars to have parts with composite materials . Describing several structural parts, from the car body, BMW shows the real potential of

composite materials from a vehicle. In order to obtain a low weight of the car, the model was build from the begining with reinforced carbon polymers in the car body and the internal structure. They have adopted the use of polymers reinforced with carbon for both the passenger segment on the model i3 and also for the i8, the main reson being that the material is as resistant as steel but at half of its weight leading thus to a very good acceleration, improved handling and uncontestable efficency [WWW_5] [WWW_7].

In tabel 1 are presented composite components based on natural fibers used for vehicle from different producers.

Tabel 1. Composite materials based on natural fibers used by different types of manufactureres from car industry

Producers Models Composite components based on natural fibers

Ford Mondeo, Focus Door panel, B column, trunk elements

Audi A2, A3, A4, Avant, A6 Seats back part, back and side door panels, spare tire comparment

Toyota Brevis, Harrier, Celsior, Raum

Door panels,backrests of the seats, spare tire lid

Mercedes Benz Trucks Engine lid, engine isolation, sun protector, front bumper

protection,cover,interior isolation BMW Seria 3, 5, 7 Door panels, phonic

isolation,backrests,tire protections VW Golf, Passat, Variant,

Bora, Polo, Fox Door panels,backrests of the seats,

trunk elements

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Ch.2. Research of elastic properties and resistance of composite materials

2.1. Introduction Phisico-mechanical properties of composite materials depend on a lot of factors the most

important being the following: - physico-mechanical properties of the constituent materials; - volumic proportions of the constituents; - spatial arrangement of the constituents; - adhesion between the constituent materials on the separation surfaces; - lenght of the fibers used; -the technological procedure for obtaining materials(pressures,temperatures,additional

materials); Besides these factors, the mechanical behaviour of composite materials is influenced by

the outdoor enviromental factors (humidity,temperatures,radiations,chemical agents), by mechanical loads on which they are subjected to (type of load,time variation of the load, speed of loading,direction of loading, time of loading).

2.2. Theoretical considerations We consider a straight bar of lenght l , with a rectangular cross section, with the width b

and thickness h , formed from n strata of composite materials with constant thickness. We consider that in section jxx = there is a variation of mechanical properties of the bars material (fig.2.1).

Fig. 2.1

In order to assure the continuity of tensions and deformations on the separation surfaces of the layers it is considered that the variation of properties is not done suddenly in the section of abscissa jx but is realized on a part from its vecinity, a part of lenght ju2 . Thus we obtain the relation for the elasticity modulus:

∑∑

+

=

=

=1

1

1

p

jn

kkjk

j

Eh

lh

lE (2.14)

In which kjE is the elasticity modulusof the material of layer k between the abscissa sections 1−jx and jx , where 1−−= jjj xxl .

For a better understanding we consider the following types of bars: - type 1 bar formed from n identical layers from a dimensional,elastic a resistance properties point of view. We will note sE elasticity modulus, )(s

rσ failiure resistance and with )(srε

elongation at failiure, for one layer of the bar. - type 2 which differs from type 1 by the fact that t between the layers have between the abscissa sections 1x şi 2x an area formed only by resin, the reinforcement being removed. We note with mE elasticity modulus, )(m

rσ failiure resistance and with )(mrε failiure elongation, for

the resin material.

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Type 2 bars can be considered as bars with different ununiformities that appear due to defects in the technological proces of production. We define the folloswing factors that characterize quality:

- Elasticity factor ideal

E EEf = (2.26)

In which E is the elasticity modulus of the material from the test bar studied, and idealE is the elasticity modulus of the real material considered ideal, without any ununiformities

- resistance factor idealr

rfσσ

σ = (2.28)

In which rσ is the failiure resistance of the material from the test bar, and idealrσ is the resistance at failiure for the ideal material without any ununiformities. We obtain:

- Ununiformity factor E

u fff σ= (2.30)

The three factors give informations for the properties of test bar materials studied, properties that are compared with that of the reference material that is considered ideal. Values that are close to 1 for the three mentioned factors indicate the fact that the studied material has very similar properties to reference one. A loss in value for the three factors indicate the presence of production defects or the fact that the studied material is different from the one that is studied.

Because the elasticity modulus for the resin has a much more smaller value than that of the elasticity modulus for the layers, by neglecting it we obtain the following simplified relations: - for the elasticity modulus of the bar

)()()(

12 xxtltnEtnl

E s

−+−−

= (2.32)

- for the failiure resistance of the bar

ntn s

rr

)()( σσ

−= (2.33)

- for the elasticity factors

)()()(

12 xxtltntnlf E −+−

−= (2.34)

- for the resistance factor

ntnf −

=σ (2.35)

- for the ununiformity factor

nlxxtltnfu

)()( 12 −+−= (2.36)

2.3. Experimental determinations 2.3.1. Used aparatus Several traction tests where used for different types of composite materials. The

apparatus used is a universal one, being adapted for static tests and for dynamic test, having a capacity of 300 kN and is used for static tests of bending,compression,traction,shear,fatigue and other mechanical test in accordance to the international norms.

Photographs for the sections where done with the electronic microscope (SEM – Scanning Electron Microscope) having a conventional catode mainly for microscopic studies of

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structure and of material surfaces, with the possibility of determining the chemical composition and of the phases in the structure.

2.3.2. Studies of ununiformities for composite materials reinforced with synthetic fibers

One of the sets of composite materials studied was of epoxy resin Resoltech 1050 with a hardener Resoltech 1058 S. On the results provided by subjecting the test bars at a traction load the following results where obtained: - failiure resistance 53-56 MPa, - failiure elongation 3.3-3.5 %, - elasticity modulus 2890-2970 MPa.

In fig 2.5 is presented a characteristic curve for a test bar made from epoxy resin.

Fig. 2.5. Characteristic curve of a test bar made from epoxy resin

From this epoxy resin I have realized sets of test bars by reinforcing them with synthetic fibers made of carbon,kevlar and glass. Based on the number of interrupted layers and of the lenght of interruption, we adopt the following symbols : - 0-0 for a reference set in which the test bars are formed by five continous layers; - 1-0 for a refernce set with five layers from which one of the layers has an interruption of zero centimeters; - 1-2 for a reference set with five layers from which one of the layers has an interruption of two centimeters; - 1-4 for a reference set with five layers from which one of the layers has an interruption of four centimeters; - 2-0 for a reference set with five layers from which two of the layers have an interruption of zero centimeters; - 2-2 for a reference set with five layers from which two of the layers have a interruption of two centimeters; - 2-4 for a reference set with five layers from which two of the layers have a interruption of four centimeters. In fig 2.6 . is presented a characteristic curve for a representative test bar from the reference set of composite reinforced materials with carbon fiber.

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Fig. 2.6. Characteristic curve for a representative test bar from the reference set of composite

reinforced materials with carbon fiber . Tabel 2.1. Experimental results and quality factors for composite reinforced material with carbon fiber

Test bar Elasticity modulus (MPa)

Failiure resistance

(MPa)

Elongation failiure

(%)

Elasticity factors

Resistance factors

Uniformity factors

0-0 39057 456 1.18 1 1 1 1-0 38734 354 0.92 0.991 0.774 0.781 1-2 37639 350 0.94 0.963 0.776 0.805 1-4 36052 356 0.98 0.923 0.780 0.845 2-0 38526 274 0.70 0.986 0.600 0.608 2-2 35469 270 0.76 0.908 0.592 0.651 2-4 32465 267 0.82 0.831 0.585 0.703

Tabel 2.2 Experimentals results and quality factors for composite materials reinforced with carbon and kevlar fiber.


Failiure resistance

(MPa)

Elongation failiure

(%)

Elasticity factors

Resistance factors

Uniformity factors

0-0 27034 383 1.44 1 1 1 1-0 26931 310 1.19 0.996 0.809 0.812 1-2 26014 303 1.22 0.962 0.791 0.823 1-4 25666 306 1.24 0.949 0.798 0.841 2-0 26622 231 0.88 0.984 0.603 0.612 2-2 24652 229 0.95 0.911 0.597 0.655 2-4 22985 234 1.05 0.850 0.610 0.717

Tabel 2.3. Experimentals results and quality factors for composite materials reinforced with kevlar fiber.


Failiure resistance

(MPa)

Elongation failiure

(%)

Elasticity factors

Resistance factors

Uniformity factors

0-0 21716 392 2.12 1 1 1 1-0 21620 316 1.71 0.995 0.806 0.810 1-2 20760 311 1.76 0.955 0.793 0.830 1-4 20021 315 1.82 0.921 0.803 0.871 2-0 21411 240 1.32 0.985 0.612 0.621 2-2 19348 238 1.41 0.890 0.607 0.682 2-4 17911 234 1.52 0.824 0.596 0.723

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Tabel 2.4. Experimentals results and quality factors for composite materials reinforced with glass fiber.


Failiure resistance

(MPa)

Elongation failiure

(%)

Elasticity factors

Resistance factors

Uniformity factors

0-0 16780 330 3.61 1 1 1 1-0 16520 268 2.92 0.984 0.812 0.825 1-2 15938 260 3.01 0.949 0.788 0.830 1-4 15322 263 3.10 0.913 0.796 0.871 2-0 16430 200 2.15 0.979 0.606 0.618 2-2 15111 195 2.30 0.901 0.591 0.656 2-4 14048 193 2.48 0.837 0.585 0.699

2.3.3. Experimental determinations on composite materials based on

dammar I have realized test bars that where made from natural resin Dammar. I have realized

three combinations of epoxy natural resin in the following way: - in the first combination 85% represents Dammar, and 15% is epoxy resin; - in the second combination 75% represents Dammar, and 25% is epoxy resin; - in the third combination 65% represents Dammar, iar 35% is epoxy resin;

Based on subjecting the test bars at traction loading I have obtained the following results. Main mechanical properties obtained for test bars with 85% Dammar and 15% epoxy

resin are: - failiure resistance 11-13 MPa, - elongation failiure 1.7-1.9%, - elasticity modulus 820-965 MPa.

Main mechanical properties obtained for test bars with 75% Dammar and 25% epoxy resin are: ` - failiure resistance 22-24 MPa, - elongation failiure 1.6-2.2%, - elasticity modulus 1220-1335 MPa.

Main mechanical properties obtained for test bars with 65% Dammar and 35% epoxy resin are: ` - failiure resistance 26-29.5 MPa, - elongation failiure 1.4-1.9%, - elasticity modulus 1740-1913 MPa.

From the resin with 75% Dammar and 25% epoxy resin I have realized a set of test bars by reinforcing them with natural fibers such as cotton,in,silk. First set of tests bars was reinforced from a mix of 40% cotton and 60% in, the mixture having a specific weight of

2/240 mg . Main mechanical properties obtained for test bars made of the above mentioned material

are: ` - failiure resistance 71-74 MPa, - elongation failiure 3.2-3.5 %, - elasticity modulus 5072-5215 MPa.

A second set of test bars was reinforced with a mixture of 40% cotton and 60% silk, the mixture having a specific weight of 2/160 mg . Main mechanical properties obtained for test bars made of the above mentioned material are: ` - failiure resistance 41-43 MPa, - elongation failiure 9,6-10.8 %,

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- elasticity modulus 2425-2556 MPa. A third set of test bars was reinforced with cotton, having a specific weight of 2/130 mg . Main mechanical properties obtained for test bars made of the above mentioned material

are: ` - failiure resistance 65-68 MPa, - elongation failiure 8.7-9.5 %, - elasticity modulus 3297-3415 MPa.

A fourth set of test bars was reinforced with hemp, having a specific weight of 2/350 mg .

Main mechanical properties obtained for test bars made of the above mentioned material are: ` - failiure resistance 73-75 MPa, - elongation failiure 2.3-2.4 %, - elasticity modulus 6410-6687 MPa.

2.3.4. Study of non-uniformities for composite materials reinforced with natural fibers

For the study of ununiformities of composite materials reinforced with natural fibers, I have realized four sets of test bars. First set was realized from epoxy resin reinforced with hemp.

\ Fig. 2.23. Characteristic curve for a representative test bar from the reference set of composite

material reinforced with hemp

Tabel 2.7. . Experimental results and quality factors for composite material reinforced with hemp Test bar Elasticity

modulus (MPa)

Failiure resistance

(MPa)

Elongation failiure

(%)

Elasticity factors

Resistance factors

Uniformity factors

0-0 6050 90 2.3 1 1 1 2-0 6033 84 2.2 0.997 0.933 0.935 2-2 5792 80 2.0 0.956 0.888 0.928 2-4 4795 69 2.4 0.792 0.766 0.967 3-0 5998 73 1,8 0.991 0.811 0.818 3-2 5623 69 1.7 0.929 0.766 0.824 3-4 4197 52 2.0 0.694 0.577 0.831

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The second set of test bars was realized from epoxy resin reinforced with cotton. Tabel 2.8. Experimental results and quality factors for composite material reinforced with cotton


Failiure resistance

(MPa)

Elongation failiure

(%)

Elasticity factors

Resistance factors

Uniformity factors

0-0 5994 88 7.2 1 1 1 4-0 5575 71 4.8 0.930 0.806 0.866 4-2 4416 45 5.5 0.736 0.511 0.694 4-4 4174 40 2.3 0.696 0.454 0.652 6-0 4498 44 3.6 0.750 0.500 0.667 6-2 4224 37 1.8 0.704 0.420 0.596 6-4 4132 33 1.3 0.689 0.375 0.544

A third set was realized from a combination of 75% Dammar and 25% epoxy resin reinforeced with hemp. Tabel 2.9. Experimental results and quality factors for composite material from a combination of 75% Dammar and 25% epoxy resin reinforeced with hemp.


Failiure resistance

(MPa)

Elongation failiure

(%)

Elasticity factors

Resistance factors

Uniformity factors

0-0 4569 56 2.4 1 1 1 1-0 4497 47 1.9 0.984 0.839 0.852 1-2 4284 48 2,0 0.937 0.857 0.914 1-4 4090 47 2.2 0.895 0.839 0.937 2-0 4488 40 1.4 0.982 0.714 0.727 2-2 3973 39 1.7 0.869 0.696 0.801 2-4 3410 39 1.8 0.746 0.696 0.932

A fourth set was realized from a combination of 75% Dammar and 25% epoxy resin reinforeced with cotton. Tabel 2.10.Experimental results and quality factors for composite material from a combination of 75% Dammar and 25% epoxy resin reinforeced with cotton.


Failiure resistance

(MPa)

Elongation failiure

(%)

Elasticity factors

Resistance factors

Uniformity factors

0-0 2875 44 3.3 1 1 1 2-0(1-0) 2708 35 2.5 0.942 0.795 0.843 2-2(1-2) 2624 35 3.0 0.912 0.795 0.871 2-4(1-4) 2483 35 3.2 0.863 0.795 0.921 4-0(2-0) 2670 31 2.1 0.928 0.704 0.758 4-2(2-2) 2352 32 2.4 0.818 0.727 0.888 4-4(2-4) 2197 30 2.4 0.764 0.681 0.891

2.4. Conclusions The properties of the studied bars depend on:

- properties of the constituent materials (elasticity modulus sE and failiure resistance )(srσ , of

the reference material, respectively the elasticity modulus mE of the matrix);

- number of the interrupted layers ( ratio nt between the number of interrupted layers and total

number of layers);

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- the lenght of interrupted layers (ratio l

xx 12 − between the interruption lenght and the lenght of

the bar). The experimental results obtained for the elasiticity modulus for reinforced carbon bars,

the carbon-kevlar, kevlar and glass bars allow the calculations of the factors that characterize the quality of the simplified relations (2.34), (2.35) şi (2.36). Tabel 2.11. The elasticity factor for composite epoxy resin materials reinforced with carbon fibers,carbon and kevlar,kevlar and glass.

Test bar Theoretical

Experimental Carbon Carbon şi

kevlar Kevlar Glass

1-0 1 0.991 0.996 0.995 0.984 1-2 0.960 0.963 0.962 0.955 0.949 1-4 0.924 0.923 0.949 0.921 0.913 2-0 1 0.986 0.984 0.985 0.979 2-2 0.900 0.908 0.911 0.890 0.901 2-4 0.818 0.831 0.850 0.824 0.837

Tabel 2.12. The resistance factor for composite epoxy resin materials reinforced with carbon fibers,carbon and kevlar,kevlar and glass.



kevlar Kevlar Glass

1-0 0.800

0.774 0.809 0.806 0.812 1-2 0.776 0.791 0.793 0.788 1-4 0.780 0.798 0.803 0.796 2-0

0.600 0.600 0.603 0.612 0.606

2-2 0.592 0.597 0.607 0.591 2-4 0.585 0.610 0.596 0.585

Tabel 2.13. The uniformity factor for composite epoxy resin materials reinforced with carbon fibers,carbon and kevlar,kevlar and glass.



kevlar Kevlar Glass

1-0 0.800 0.781 0.812 0.810 0.825 1-2 0.833 0.805 0.823 0.830 0.830 1-4 0.865 0.845 0.841 0.871 0.871 2-0 0.600 0.608 0.612 0.621 0.618 2-2 0.667 0.651 0.655 0.682 0.656 2-4 0.733 0.703 0.717 0.723 0.699

From the analysis of tabel 2.11, 2.12 and 2.13 the following conclusions can be stated: - the elasticity factor decreases if the number of interrupted layers grows; - the elasticity factor decreases if the lenght of interrupted layers increases; - the resistance factor decreases if the number of interrupted layers increases; - the resistance factor is not influenced by the lenght of the interrupedt layers; - the uniformity factor decreases if the number of interrupted layers increases; - the elasticity factor increases if the lenght of the interruped layers increases;

16

Ch.3. Vibration research upon slender composite bars

3.1. Introduction Bars are structural elements often used in common practice and are studied the most by

the Euler-Bernoulli model, which is based on the hypotheses that a normal plane section of the fiber before deformation remains normal plane on the entire duration of the deformation. The Euler-Bernoulli theory underestimates the deformations and overestimates the natural frequencies due to the fact that it ignores the transversal deformation due to shearing. The Euler-Bernoulli hypotheses are valid only for bars that have the transversal section very small with respect to bars lenght.

The main hypotheses used for the studied models are: - a displacement similar to the one from simple bending; - the displacement component due to deformation by shearing can vary in section

according to different laws (parabolic,sinusoidal,hyperbolic or exponential). - the transversal displacement is thought to be a function only of the longitudinal variable

of the bar; - a constitutive unidimensional equation is used; - the bars are subjected only to exterior transversal loads; - during the deformation no other suplementary support appear. 3.2. Movemenet equation We consider a bar with constant cross section, of rectangular shape of width b2 and

thickness h2 (as in figure 3.1).

Figura 3.1.

To the cross section we have a reference system: - axis x is the longitudinal symetric axis of the bar; - axis y is orientated on the thickness; - axis z is oriented on the direction width (as in figure 3.1).

If we consider only the transversal vibrations, neglecting the longitudinal and torsional vibrations, due to symmetry the functions that characterize the displacements according to the three axis have to fulfill the next conditions: - displacement xw has to be an even function in variable z and odd in variable y , thus

( ) ( );;;;;;; tzyxwtzyxw xx =− (3.4) ( ) ( );;;;;;; tzyxwtzyxw xx −=− (3.5)

- displacement yw has to be an even function in variable z and also in variable y , thus ( ) ( );;;;;;; tzyxwtzyxw yy =− (3.6)

17

( ) ( );;;;;;; tzyxwtzyxw yy =− (3.7) - displacement zw has to be an even function in variable z and also in variable y , thus

( ) ( );;;;;;; tzyxwtzyxw zz −=− (3.8) ( ) ( )tzyxwtzyxw zz ;;;;;; −=− . (3.9)

In order to fulfill these conditions the following state of displacement is considered: ( ) ( )( ) ( )

( ) ( ).;;;;

;;;;;;;;;;

2

3

2

2

tyxbztyxzw

tyxvztyxuwtyxvztyxuw

z

yyy

xxx

ϕθ +=

+=

+=

(3.10)

The functions that characterize the bars deformations are considered to be: ( ) ( ) ( ) ( )yatxwyutxwux ;''';' += , (3.24.1) ( ) ( ) ( ) ( )ybtxwyvtxwuy ;''; += , (3.24.2) ( ) ( ) ( ) ( )yptxwyftxw ;''; +=θ , (3.24.3) ( ) ( ) ( ) ( )yqtxwygtxw ;''; +=ϕ . (3.24.4)

Functions ( )yu , ( )ya , ( )yf , ( )yg , ( )yp and ( )yq are odd in variable y , and functions ( )yv and ( )yb are even in this variable.

We considere the above mentioned functions of the form:

( )

( )

( )

( )

( )

( ) .

,1

,

,

,

,

2

112

2

2

112

2

12

11212

12

11212

12

11212

12

11212

k

kkk

k

kkk

k

kkk

k

kkk

k

kkk

k

kkk

yhbyb

yhvyv

yhqyq

yhpyp

yhfyf

yhgyg

⋅=

⋅+=

⋅=

⋅=

⋅=

⋅=

∑

∑

∑

∑

∑

∑

=−

=−

−

=−−

−

=−−

−

=−−

−

=−−

(3.41)

In the case of Euler-Bernoulli theorem, the equation of transversal vibrations is:

( )( )

,; 2

2

yS

y pxMdStxw =

∂∂

+⋅∫∫••

ρ (3.52)

where ( ),;'' txwEIM = (3.54)

with

( ) ( ) .8031

34

51

56

34 33

−=

−= hhg

EE

EbhhvEbhEIyy

yzxxxx (3.55)

Detailed

( ) .280

913

42

23

xxzzyyzyyzyy

yzzyyz EEEEEE

EEEbhbhEI

+−= (3.56)

As well

18

( ) ( ) .3

42

+= hg

hbhvbhA ρρ (3.59)

Detailed

.1691

214 2

2

+

++=

yy

yz

zzyyyzzy

yzzy

EE

bh

EEEEEE

bhA ρρ (3.60)

If it is considered 0=υ , results:

,4

,3

4 3

bhA

EbhEI

ρρ =

= (3.62)

Likewise in the clasical Euler-Bernoulli theorem. With EI and Aρ previous introduced, the equation of transversal vibrations for the

bar will have the form:

( ) ( ) ,;''''; yptxwEItxwA =+••

ρ (3.63) Which is identical to the classical Euler-Bernoulli equation for the study of transversal

vibrations on bars. The free vibrations of the bars are of the form: )sin()(),( nn

nn txWtxw ϕω += ∑ (3.64)

In which )(xWn are its own functions and depend on the limit conditions of the bar. The string of its own pulsation is given by the relation:

,2

2

AEI

Ln

n ρβ

ω = (3.65)

In which nβ depens on the limit conditions of the bar. If the transversal contraction coefficient are taken into account the own vibrations have a smaller value than the case in which they are neglected. Because for a given thickness of the bar, when the width is incereased the

ratio A

EIρ

increases, resulting in a increase of own pulsation with the increase of the bars width.

3.3. Damped free vibrations Vibrations are called to be free if the exterior loading is null. In this case the vibrations are due to the initial conditions. The movement equation will be of the form:

( ) ( ) 0;''''; 2 =+••

txwatxw (3.66) In which

AEI

aρ

= (3.67)

In reality due to internal frictions and interaction with air,all vibrations are damped. The presence of mechanism that disipate energy is now accepted in all models used for the study of mechanical vibrations in mechanical systems. A detailed study presented in [HER_2008], by

introducing in the movement law some terms like •

wc02 or 2

2

12x

wc∂∂

−

•

or 4

4

22x

wc∂∂

•

. The term

•

wc02 introduces the so called external or viscous damping. The amplitude of all the vibrations modes (modal amplitudes) are damped with the same rate in contrast with experience. A natural

19

interpretation of the term 2

2

12x

wc∂∂

−

•

is that the damping force is directly proportional to the

bending speed. The presence of the term 4

4

22x

wc∂∂

•

mean that the damping rates of the own

modes of vibration depend proportionaly to the squared frequency. This is the so called Kelvin-Voigt model for internal damping. 3.3.1. Case of the simply supported bar Therefore in the presence of damping the movement equation is:

( ) 0);();(2);(2);(2; 4

42

4

4

22

2

10 =∂

∂+

∂∂

+∂

∂−+

•••••

xtxwa

xtxwc

xtxwctxwctxw (3.68)

The bar being simply supported at its end the limit conditions are: 0);();0( == tLwtw (3.69)

Respectively

0);();0(2

2

2

2

=∂

∂=

∂∂

xtLw

xtw (3.70)

where L is the length of the bar. The limit conditions allow in relation (68) the usage of the finite Fourier transformed into sinus.

The vibration of the bar is obtained by inverting the transformed Fourier:

Lxntft

fge

L

Lxntft

fge

Ltxw

nnnnnnn

nn

nnnt

n

nnnnnn

nn

nnnt

n

n

πωµωµωµ

µ

πµωµωµω

µ

µ

µ

sin)cosh()sinh(2

sin)cos()sin(2),(

0

0

2222

22

1

1

2222

22

∑

∑

∞

=

−

−

=

−

−+−

−

++

+

−+−

−

+=

(3.85)

It is noticed that the own string pulsations depend only on the elastic properties of the bars material and of the dimensions of the transversal section. In the case of undamped vibrations the own pulsation are inversely proportional to the bars squared lenght. In the case of damped vibrations, due to the damping mechanisms, there is noticed a decrease of the pulsation vibrations. It is noticed that the bars lenght has a great influence upon damping, function of the type of damping, the damping factor being constant, inversly proportional to the bars squared lenght or inversly proportional to the bars lenght at the power four. 3.3.2 Method of variable separation In general there are studied the damping due to air friction and internal damping of the bars material. In this case the movement equation has a particular form:

( ) 0);();(2);(2; 4

42

4

4

20 =∂

∂+

∂∂

++

••••

xtxwa

xtxwctxwctxw (3.89)

The equation is solved in the initial conditions given by relations (3.71) şi (3.72) and of different limit conditions. The solution of the equation is given as a multiplication of two functions:

)()(),( tTxWtxw = (3.90) function )(xW depends only on variable x , and function )(tT depends only of time. Replacing (3.89) it can be given in the form:

20

)()(2

)(2)()(

)(''''2

2

0

tTatTc

tTctTxW

xW

+

+−= •

••

(3.91)

In relation (3.91) the term on the left side of the equality is function only of the variable x , and the term on the right side is function only with respect to time. The equality is realized only if the two terms are equal to a constant, noted as 4λ for a simplifyed calculus. Thus:

4

)()('''' λ=

xWxW (3.92)

4

22

0

)()(2

)(2)(λ=

+

+− •

•••

tTatTc

tTctT (3.93)

The general solution (3.92) is: )cos()sin()cosh()sinh()( 4321 xCxCxCxCxW λλλλ +++= (3.94)

Constants 4321 ,,, CCCC are determined from end conditions . Therefore the vibrations of the bar will be:

)()cosh()sinh(

)()cos()sin(),(

0

0

2222

22

1

1

2222

22

xWtftfg

e

xWtftfg

etxw

nnn

nnnnn

nn

nnnt

n

n

nnnnnn

nn

nnnt

n

n

∑

∑

∞

=

−

−

=

−

−+−

−

++

+

−+−

−

+=

ωµωµωµ

µ

µωµωµω

µ

µ

µ

(3.110)

In general, in the case of damping, the free vibrations of the bar will be of form(3.110),

each of the own vibrations modes having its own damping factor. Term •

wc02 leads to a constant

damping factor, the term 2

2

12x

wc∂∂

−

•

leads to an inversely proportional damping factor with

respect to the bars squared lenght, and in the presence of the term 4

4

22x

wc∂∂

•

the damping factor

is inversely proportional to the bars length to power four. 3.4. Experimental determinations I have realized test bars having Dammar natural resin. Materials based only on this type of resin have a long hardening period. In order to reduce the time I have used a small proportion of synthetic resin. More precisely I have used 75% dammar and 25% epoxy resin. From this type of resin I have realized four sets of composite materials by reinforcing with in,cotton,silk and hemp. I have realized a first set of bars from this resin and reinforced it with: -mixture of 40 % cotton and 60 % in, havin specific mass 2/240 mg (these test bars will be called reinforced with in). - mixture of 60 % silk and 40 % cotton, with specific mass (these test bars will be called reinforced with silk). -mix of cotton, with specific mass 2/130 mg . - mix of hemp, with specific mass 2/350 mg .

AmI have determined experimentaly the damping coeffcient for these sets of test bars. The studied bars had lenghts of 200 mm and widhts of 10 mm, 15 mm, 20 mm and had one fixed end, the vibrations measurement being performed at the free end. The free lenght of the bars end was of 100 mm, 120 mm, 140 mm,160 mm and 180 mm.

The used measurement aparatus was:

21

- accelerometer with sensibility of 0.04 2/ −mspC ; - recording data system SPIDER 8; - signal conditioner NEXUS 2692-A-0I4 connected to SPIDER 8. The set of recorded data was done with the software CATMAN EASY, and the frequency domain was 0 - 2.400 Hz from SPIDER 8.

In figure 3.4. is presented the way of determining the damping factor for the record shown in figure 3.3.The determination of the damping factor for the unight weight of the bar was done with the relation:

,ln1

2

1

12 ww

tt −=µ (3.115)

- 1t and 2t are the times for which two maximums are obtained on the experimental diagram; - 1w is the maximum value at time 1t , and 2w is the maximum value obtained at time 2t

Fig. 3.4. Determining the damping factor for a test bar renforced with hemp having width 20

mm and free lenght of 100 mm In tabels 3.1-3.4 are the values of the damping factor and measured frequencies, measured experimentaly for the four sets of test bars. Tabel 3.1. Damping factor and frequency for bars reinforced with cotton (experimental records

are presented in annex 3.1.) (thickness of test bars: 5.6 mm) Width

Lenght

10 mm 15 mm 20 mm Damping

factor [s-1]

Frequency [Hz]

Damping factor [s-1]

Frequency [Hz]


Frequency [Hz]

100 mm 23.99 81.35 21.24 91.25 21.16 96.38 120 mm 15.76 58.25 16.99 65.04 17.98 70.37 140 mm 12.33 43.02 13.89 51.28 13.77 54.05 160 mm 9.54 34.04 9.74 40.61 11.28 41.73 180 mm 7.57 28.80 8.88 31.29 9.39 34.08

Tabel 3.2. Damping factor and frequency for bars reinforced with in experimental records are

presented in annex 3.2.) (test bar thickness: 5.9 mm) Width

Lenght


factor [s-1]

Frequency [Hz]


Frequency [Hz]


Frequency [Hz]

100 mm 25.43 112.15 30.33 118.81 35.89 122.45 120 mm 18.87 81.63 22.83 87.59 29.80 89.55 140 mm 14.29 61.38 17.60 67.79 21.94 70.18 160 mm 10.55 46.87 16.39 50.20 15.61 53.81 180 mm 8.77 38.09 11.32 40.13 11.87 42.40

22

I have realized the second set of test bars from this combined resin and reinforced it with: - 6 layers of in, having in the middle a layer of fiber glass (thickness of the bars 3.3 mm), - 12 layers of cotton, having in the middle a layer of fiber glass (thickness of test bars of

3.2 mm). Tabel 3.5. Damping factor and frequency for bars reinforced with in and a layer of fiber glass

(experimental records are presented in annex 3.5.) Lăţime

Lungime


factor [s-1]

Frequency [Hz]


Frequency [Hz]


Frequency [Hz]

100 17.27 77.86 17.72 88.16 17.91 100.42 120 12.68 57.62 14.23 66.53 11.98 70.43 140 11.27 43.40 11.58 50.20 10.12 55.52 160 9.34 35.10 9.45 40.37 7.12 42.18

Tabel 3.6. Damping factor and frequency for bars reinforced with cotton and a layer of fiber glass (experimental records are presented in annex 3.6

Width

Lenght


factor [s-1]

Frequency [Hz]


Frequency [Hz]


Frequency [Hz]

100 14,79 66,39 13,10 75,16 15,78 85,33 120 12,18 47,36 12,44 53,01 10,96 65,41 140 9,87 36,44 9,62 41,27 9,70 47,03 160 7,89 29,06 6,75 31,14 6,23 34,33 I have realized a third set of test bars form this combined resin and reinforced it with:

- 6 layers of in, having in the middle two layers of fiber glass (thickness of test bars is of 3.6 mm),

- 12 layers of cotton, having in the middle two layers of fiber glass (thickness of the test bars is of 3.5 mm). The studied test bars had lenghts of 200 mm and widths of 10 mm, 20 mm, 30 mm and where fixed at one end, the measurement of the vibrations being performed at the free end. Free lenght for each bar was 100 mm, 120 mm, 140 mm and 160 mm.

In tabel 3.7 are presented values determined experimentaly for the damping factor and frequency of the first vibration mode for the test bars of the third set of test bars reinforced with in.

Tabel 3.7. Damping factor and frequency for the test bars reinforced with in and two layers of

fiber glass (experimental records are presented in annex 3.7.) Width

Lenght


factor [s-1]

Frequency [Hz]


Frequency [Hz]


Frequency [Hz]

100 19.75 110.20 20.66 125.16 18.36 139.94 120 16.85 80.14 14.36 89.88 15.28 99.58 140 12.02 61.67 13.35 66.99 14.26 72.89 160 9.65 47.13 11.45 50.61 10.90 55.11

In tabel 3.8 are presented values determined experimentaly for the damping factor and

frequency of the first vibration mode for the test bars of the third set of test bars reinforced with cotton.

23

Tabel 3.8. Damping factor and frequency for the test bars reinforced with cotton and two layers of fiber glass (experimental records are presented in annex 3.8.)

Width

Lenght


factor [s-1]

Frequency [Hz]


Frequency [Hz]


Frequency [Hz]

100 19,06 91,77 18,15 112,68 21,82 118,96 120 15,61 60,95 13,08 79,93 14,15 84,95 140 10,95 46,48 10,44 60,34 9,46 63,70 160 8,47 39,72 7,26 46,65 8,32 46,04

3.5. Discussions and conclusions

The theory proposed is based on an asymmetric distribution of deformations with respect to the median plane, the zone subjected to compressions suffering a transversal dilatation and the zone subjected to streching is contracting transversaly. In consequence the points from a croos section of the bar will no longer have the same transversal displacement. Results thus an increase of own pulsation with the increase bar width. This increase was experimentaly checked for the four test bars with composite matrix based on natural Dammar resin, reinforced with cotton,in,silk, and hemp. A conclusion regarding the influence of bar width on damping properties can not be drawn. Thus in the case of test bars reinforced with cotton it has been observed a small increase of damping factor with respect to the bars width and for the test bars reinforced with hemp the phenomenom is the other way around, the damping coeffcient decreasing with the bar width.

Results that for every bar lenght the ratio 2

2

ρυEh is constant.

In tabel 3.9 are presented value of this ratio computed with frequency values measured for bars having the free lenght of 100 mm.

Tabel 3.9. Variation of the ratio 2

2

ρυEh for test bars reinforced with

In,cotton,silk or hemp (in 4m ) Width Reinforced materials

Cotton In Silk Hemp 10 mm 0.013714 0.012164 0.013771 0.013670 15 mm 0.010899 0.010628 0.010339 0.010801 20 mm 0.009770 0.010005 0.009670 0.010076

Similar conclusion can be withdrawn from the test bars that have in the middle one or more layers of fiber glass.

24

Ch.4. Research on vibrations for multilayer composite bars

4.1. Introduction A special place in the study of dynamics of composite material is the one of sandwich bars

formed from several overlapped layers with constant thickness . Most studies reffer to sandwich bars made out of three layers, the middle one having a viscous-elastic behaviour, the inferior and superior layers having elasting and resistance properties. Most studies are based on the following assumptions regarding the behaviour of laminated sandwich bars:

- there is a continuity between the displacements and tensions on the separation surfaces of the layers;

- mainly transversal inertia forces govern, neglecting the longitudinal inertia and rotation inertia of the bars cross section;

- there are no deformations on the thickness of the bar, thus transversal deformations are the same on the entire cross section of the bar;

- the core has an elastic or viscous-elastic behavoiur taking the shearing tensions (tangential ones);

- the core is not subjected to normal tensions; - the exterior layers have an elastic behaviour, being subjected to pure bending. 4.2. Theoretical considerations A sandwich bar is considered of width b , having symmetrical geometry with respect to the

middle layer (figure 4.1).

Fig. 4.1

We note with 1,1, −= nkyk , the abscisa between the separation layers k and 1+k and with ny the abscisa of the exterior layer n . Due to symmetry we highlight the deformations of the base for 0≥y . We consider that the displacements after axis Ox and Oy , for layer k , are of the form:

( )( ) ( ) ( ) ( )

( )( ) ( ) ,,1,,,,

,,,,, 3

nktxwtyxux

txwytxyytyxu

ky

kk

x

==∂

∂⋅−⋅⋅= αB

(4.1)

Bending moment that acts in the cross section of the bar is:

,2

2

21 xwEI

xEIM

∂∂⋅+

∂∂⋅−=α (4.7)

where

25

.3

2

,53

2

31

3

12

51

531

3

11

−

=

−−

=

−=

−−=

∑

∑

kkn

kk

kkkkk

n

kk

yyEbEI

yyyyEbEI B (4.8)

Shear force that acts in the cross section of the bar is : ,αGAT = (4.9)

where

( ) .2 31

31

1kkkkk

n

kk yyyyGbGA B−−

=

−−= ∑ (4.10)

The movement equations for the transversal vibrations of the bar in xOy plane, are:

( ),0=

∂∂

−−∫∫••

xTpdSu y

Sy ρ (4.11)

( ),0=

∂∂

++∫∫••

xMTdSuy

Sx ρ (4,12)

where ( )yx,ρ is density, and yp is exterior charge. Movement equation will be of the form:

,0=∂∂

−−••

xGApwA y

αρ (4.13)

,03

3

22

2

121 =∂∂

+∂∂

−+∂∂

−••

••

xwEI

xEIGA

xwII ααραρ (4.14)

In which

( )

,3

2

,53

2

,2

31

3

12

51

531

3

11

11

−

=

−−

=

=−

−=

−−=

−=

∑

∑

∑

kkn

kk

kkkkk

n

kk

n

kkkk

yybI

yyyybI

yybA

ρρ

ρρ

ρρ

B (4.15)

where kρ is density of layer k . We note

- Medium density: ( )∑=

−−==n

kkkk yy

hAA

11

2 ρρ

ρ ; (4.19)

- Medium rotation: ( ) ( )( )

αθ2

1

2

,,,1EIEI

xwdStyxuyxyE

EI Sx −

∂∂

=−

= ∫∫ ; (4.20)

- Medium shearing modulus: ( )( )

( )∑∫∫=

−−==n

kkkk

S

yyGh

dSyxGA

G1

12,1 ; (4.21)

- Medium elasticity modulus: I

EIE 2= . (4.22)

Thus the movement equations will be:

ypxx

wKGAwA =

∂∂

−∂∂

−•• θρ 2

2

, (4.25)

respectively

26

02

2

21 =∂∂

+

−∂∂

+

−

∂∂ ••

••

xEI

xwKGAK

xwKI θθθρ , (4.26)

In which

1

2

EIEI

GAGA

K ⋅= , (4.27)

1

12211 EII

EIIEIIK

ρρρ −

= , (4.28)

1

212 EII

EIIK

ρρ

= , (4.29)

are coefficients that take into account the ununiformities of the tensions in the the cross section . In the case of a homogenous bar we have:

1,0,65

21 === KKK ,

And the movement equation are the same as from clasical Timoshenko bars.

4.3. Free vibrations The bar has free vibrations if 0=yp . In this case the solutions are to be found with the

method of sperating the variables: ( ) ( ) ( )tTxWtxw =, , (4.30) ( ) ( ) ( )tTxYtx =,θ . (4.31)

The mathematical model of transversal vibrations can be reduced to the equations: ( ) ( ) ( ) 0''' =−⋅+ xYxWaxW , (4.32) ( ) ( ) ( ) 0''' =++ xcWxbYxY , (4.33)

( ) ( ) 02 =+••

tTtT ω , (4.34) In which

EIKIKGAc

EIKGAKIb

KGa 1

22

22

,, ωρωρρω −=

−== (4.35)

and ω is the pulsation of the vibrations. We obtain:

( ) ( ) ( ) ( )[ ]xWcaxWb

xY ''''1++−= (4.36)

and ( ) ( ) ( ) 0'''''' =++ xeWxdWxW , (4.37)

where

−

+=E

KKKG

d 122 1ρω , (4.38)

( )KGAKIKGEI

e −= 22

2

ωρρω . (4.39)

The mathematical model is identical to the one studied in depth in [MAJ_2009]. The characteristical equation attached to the differential one (4.37) is:

024 =++ edrr (4.46)

27

In which solutions 2211 ,,, λλλλ −− with 2

42

1edd −+−

=λ and

242

2edd −−−

=λ .

We note

2)(

xx

k

kk eexCλλ

λ−+

= ; 2

)(xx

k

kk eexSλλ

λ−−

= ; 2;1=k (4.47)

With this notation the solution of the differential equation (4.37) is of the form: )()()()()( 24231211 xSAxCAxSAxCAxW λλλλ +++= (4.48)

If in the abcisa point 0=x the bar is simply supported:

00

3221

21

31

=+

=+

AAAAλλ

(4.50)

From where it results 01 =A and 03 =A , because 022

21 =− λλ only if 0=ω , meaning

that the bar does not vibrate. If in the abscisa point Lx = the bar si simply supported:

0)()(0)()(

232211

21

2312

=+

=+

LSALSALSALSAλλλλ

λλ (4.51)

System (4.51) has a non zero solution only if 0)()()( 21

21

22 =− LSLS λλλλ (4.52)

From where it results 0)( 1 =LS λ or 0)( 2 =LS λ . If 1λ is real from the condition 0)( 1 =LS λ it s obtained 01 =λ from which the pulsation results as:

2IKKGAρ

ω = (4.53)

If 1λ is complex from the condition 0)( 1 =LS λ it is obtained πλ nL =1 , with *Nn∈ ,

which has solutions if 2

222L

nd π≥ . Because 2λ is always coomplex we get πλ nL =2 , with

*Nn∈ , that has solution if 2

222L

nd π≤ . In both cases the pulsation string is obtained as:

2∆±

=P

nω , cu *Nn∈ (4.54)

where

−++=

ρπ

ρKGKKE

Ln

IKGAP )( 12

2

22

(4.55)

4

44

22 4

LnKGEP π

ρ−=∆ (4.56)

4.4. Case of sandwich bar We consider a bar made from three layers, a middle layer and two exterior ones. Ratio

GAGA

will have the expression:

2

1

2

1

2

1

22

21

2

122

21

2

1

2

1

1

31

32

31

yy

yy

GG

yy

yy

yy

yy

GG

GAGA

−+

−−+

−

= (4.57)

28

Ratio 1

2

EIEI

will have the expression:

−−+

−−

−

−+

=

22

21

32

31

1

2152

51

1

2132

31

2

1

32

31

32

31

2

1

1

2

1531

154

152

156

131

31

yy

yy

GGG

yy

GGG

yy

EE

yy

yy

EE

EIEI

(4.58)

If we consider two identical materials, then 21 GG = şi 21 EE = , we obtain:

32

=GAGA

, 45

1

2 =EIEI

, and 65

=K .

If the bar is considered to be made only from the material noted with 2, then 01 =y and it is obtained:

32

=GAGA

, 45

1

2 =EIEI

, and 65

=K .

If the bar is considered to be made only from the material noted with 1, then 21 yy = and is obtained:

32

=GAGA

, 45

1

2 =EIEI

, and 65

=K . All three results are identical because it represents the case

of the homogenous bar .

Ratio IIρρ 1 will have the expression:

−+

−−+

−−

−

=

2

1

2

1

2

1

22

21

32

31

1

2152

51

1

2132

31

2

1

1

131

1531

154

152

156

yy

yy

yy

yy

GGG

yy

GGG

yy

II

ρρ

ρρ

ρρ

(4.59)

If two identical materials are to be considered, then 21 GG = and 21 ρρ = , it is obtained

541 =

IIρρ

. Same result is obtained in the case 01 =y , respectively 21 yy = .

Ratio IIρρ 2 will have the expression:

2

1

2

1

2

1

32

31

32

31

2

1

2

1

1

yy

yy

yy

yy

II

−+

−+=

ρρρρ

ρρ

(4.60)

If two identical materials are to be considered, then 21 ρρ = and it is obtained 12 =IIρρ

.

Same result is obtained in the case 01 =y , respectively 21 yy = . For example it is considered the case of a bar with the following characteristics:

mb 2102 −⋅= , mh 2103 −⋅= , mL 21040 −⋅= , 12 2ρρ = ,

12 20EE = , 12 20GG = , 6.21

1EG = , 21 93458.0 yy =

It is obtained

29

219.0=GAGA

, 599.11

2 =EIEI

, 138.01 =IIρρ

, 501.02 =IIρρ

, 106542.1 ρρ = , 124298.2 GG = ,

14903.4 EE = and ununiformity coeffcients 350.0=K , 221.02 =K and 280.01 −=K . First

own pulsation, computed with (4.54), has the value 1

11 0606.1

ρω

E= . If we consider that that

bar can be studied with the simplified Euler-Bernoulli theorem and first own pulsation is

computed with (3.65), applied for the simply supported bar we obtain 1

11 0966.1

ρω

E= . It is

noticed that the difference between the two is only of three percents, thus the Euler-Bernoulli theorem is applicable on sandwich bars. Thus the calculations for the equivalent rigidity 2EI can be used with the simplifyed relation:

4

42

2n

n LAEI

βωρ

= (4.61)

In which nβ , *Nn∈ , depending on the limit conditions of the bar. 4.5. Experimental determinations I have realized 12 plates having the core of honeycomb polypropylene on the faces of which I have added one or two layers of carbon fiber, respectively glass fiber. From each plate I have cut bars of 400 mm and widths of 30 mm, 45 mm, respectively 60 mm.

(a) (b)

Fig. 4.11 Bars with the core of honeycomb polypropylene on the faces of which I have added one or two layers of carbon fiber, respectively glass fiber

The aparatus used for determening the vibrations is the one presented in Chapter 3 and the method of measurement is identical to the one presented.

Tabel 4.2. Damping factor and frequency of vibrations for a bar with a honeycomb core of 1 centimeter thickness and a layer of carbon fiber

Width Free lenght

3 cm 4.5 cm 6 cm Damping

factor [s-1]

Frequency [Hz]


Frequency [Hz]


Frequency [Hz]

200 mm 26.98 112.02 26.27 123.50 23.75 120.81 230 mm 20.17 88.64 18.21 98.66 16.73 102.54 260 mm 18.71 73.22 16.32 79.80 13.67 82.17 290 mm 16.92 54.32 15.51 55.88 12.26 65.32 320 mm 14.00 46.28 12.21 48.44 10.33 54.09 350 mm 11.96 33.40 10.11 36.28 8.80 43.50

30

Tabel 4.3 Damping factor and frequency of vibrations for a bar with a honeycomb core of 1.5 centimeter thickness and a layer of carbon fiber

Width Free lenght


factor [s-1]

Frequency [Hz]


Frequency [Hz]


Frequency [Hz]

200 mm 38.37 153.60 38.11 158.63 36.25 153.35 230 mm 31.89 128.00 31.80 128.54 32.08 124.12 260 mm 24,86 104.74 24.63 100.55 23.29 102.53 290 mm 20.07 89.38 20.08 87.83 16.25 88.70 320 mm 17.65 75.70 17.68 76.12 13.62 75.81 350 mm 14.61 53.28 13.36 65.34 13.07 58.42

Tabel 4.7. Damping factor and frequency of vibrations for a bar with a honeycomb core of 2

centimeter thickness and a layer of carbon fiber Width

Free lenght


factor [s-1]

Frequency [Hz]


Frequency [Hz]


Frequency [Hz]

200 mm 57.38 175.82 50.26 177.78 49.55 172.66 230 mm 48.23 150.94 46.14 139.53 40.26 142.02 260 mm 44.23 128.34 44.31 121.52 34.19 123.71 290 mm 37.48 113.74 32.09 111.63 27.70 111.63 320 mm 26.02 102.56 21.45 102.35 20.67 97.95 350 mm 21.06 83.77 18.82 82.33 17.13 74.41


centimeter thickness and a layer of fiber glass Width

Free lenght


factor [s-1]

Frequency [Hz]


Frequency [Hz]


Frequency [Hz]

200 mm 39.82 164.11 32.38 167.13 31.76 160.27 230 mm 31.32 141.26 27.42 137.74 24.96 133.51 260 mm 25.34 114.14 25.57 107.03 21.75 102.79 290 mm 23.32 89.46 21.83 87.99 18.82 84.73 320 mm 19.88 76.74 18.49 74.43 14.84 74.01 350 mm 14.71 59.12 13.05 60.52 12.84 60.42


centimeter thickness and a layer of fiber glass Width

Free lenght


factor [s-1]

Frequency [Hz]


Frequency [Hz]


Frequency [Hz]

200 mm 25.24 109.49 24.97 115.94 24.80 119.71 230 mm 22.23 92.41 18.89 96.19 18.15 93.20 260 mm 16.89 74.24 16.51 70.08 14.10 73.06 290 mm 14.67 59.05 15.37 58.23 12.47 60.12 320 mm 13.38 51.77 10.11 50.12 11.48 52.43 350 mm 12.19 38.02 9.44 39.52 8.44 40.12

31

4.6. Discussions and conclusions The mathematical model used for the study of vibrations in the case of multilayer composite bars makes a generalization for the Timoshenko model, taking into account the ununiformities in the cross section of the tangential tensions and also of the ununiformities of the normal tensions. If in the Timoshenko model is introduced a coeffcient that takes into account the ununiformities of the tangential tensions in the cross section, in the mathematical model are introduced three such coeffcients that take into account also the ununiformities of the normal tensions. The proposed model by the variation in cross section of the deformation tensions cand be introduced in theories such as PSDBT (parabolic shear deformation beam theory). For each considered test bar, for all the measurements, with the measured frequency the equivalent rigidity has been calculated using relation (4.61). In compliance with (4.22) the medium elasticity modulus can be determined for the bar using the relation:

63

2 1012

⋅=bhEI

E ( MPa ) (4.62)

In relation (4.62) thickness and the width of the bar are introduced in milimeters. Tabel 4.14. Equivalent rigidity and the medium elasticity modulus for bars having the exterior layers reinforced with carbon fiber

Thickness (mm)

Width of bar (mm)

1 layer 2 layers Rigidity

2EI

( 2Nm )

Equivalent elasticity modulus ( MPa )

Rigidity 2EI

( 2Nm )

Equivalent elasticity modulus ( MPa )

10 30 5,772 1884 10,079 2721 45 8,543 1812 15,280 2750 60 11,966 1953 21,055 2842

15 30 12,461 1287 19,033 1725 45 18,687 1288 26,412 1596 60 23,716 1226 35,452 1607

20 30 20,218 912 33,080 1349 45 29,537 888 47,307 1287 60 42,828 965 58,220 1208

From the analysis of tabels 4.14.-4.16. the following conclusions can be withdrawn: - equivalent rigidity is proportional to the width of the bar, the only differences can be due to the fact that when the plates are cut, the cross sections cand differ from one bar to the other; - the medium elasticity modulus decreases with the thickness of the middle layer, fact proven by having almost the same quantity of fiber in a small cross section for the bars with honeycomb of 10 mm , cross section that increases for the bars that have a honeycomb of 15 mm and of 20 mm; - the average elasticity modulus is greater for bars having exterior layers reinforced with carbon fibers than the ones reinforced with glass fibers. From the analysis of tabels 4.2.-4.13. next conclusions cand be submitted: - the damping factor increases with the thickness of the honeycomb core; - the damping factor is bigger for the bars having two exterior layers than for the ones with only one layer; - damping factor is bigger for bars having exterior layers of carbon fiber than for the ones made of glass fiber

32

Ch.5. Experimental study, modeling and virtual simulations of car part realized from composite materials

5.1. Introduction In this chapter the production of a phisical car body part was followed (the left side

fender of a Peugeot 206 made in 1998 ) from different composite materials. On the different fenders thus created, a modal analysis has been made in order to determine its own frequencies on a certain direction, and an armonic analysis with Finite Element Method and a structural stability study for the interaction of fluid-structure.

5.2. Determination of own frequnecies by experimental measurements The experimental analysis for determining the own frequencies was made on 7 fenders,

onde made of steel slab (fig.5.1), and the other 6 made from composite materials of epoxy resin reinforced with fibers of:carbon,kevlar,carbon-kevlar,glass,cotton and hemp. The ratio used for the matrix is of 100 volumetric parts of Resoltech resin 1050 with 30 volumetric parts of hardener Resoltech 1058 S. The fenders were hand made by pouring layer after layer such as:

- For the fender reinforced with carbon fiber 3 layers have been used (fig.5.2); - For the fender reinforced with carbon-kevlar fiber 3 layers have been used (fig.5.3); - For the fender reinforced with glass fiber 2 layers have been used (fig.5.4); - For the fender reiforced with cotton fiber 5 layers have been used (fig.5.5); - For the fender reinforced with hemp fiber 3 layers have been used (fig.5.6); - For the fender reinforced with kevlar 3 layers have been used (fig.5.7).

Fig.5.2. Fender made of epoxy resin and Fig.5.6 Fender made of epoxy resin reinforced reinforced with carbon fiber with hemp fiber

Two measurements were performed as cand be seen in Fig 5.8 and 5.9. The excitation was made by an impact hammer, the point subjected to the load being presented in Fig 5.10.

Fig.5.8. Positioning of the accelerometer (lateral point)

33

In figure 5.11 is represented the experimental measurement done on the lateral side for a steel fender.

Fig.5.11. Experimental measure in the lateral side

5.3. Modeling and virtual simulation of a car part made of composite

materials 5.3.1. Tridimensional model of the fender In order to obtain the tridimensional geometric model of the fender, needed in order to

have the virtual modeland for the experimental study with Finite Element Method, a scan was made of the real fender, obtaining thus the mesh (fig. 5.17).

Fig. 5.17. Scaning front fender for Peugeot 206. Obtaining the mesh.

With the help of software application which the scaner has,the mesh obtained was

converted in a STL format, having a body with 347685 points and 686812 surfaces with triangular shape that were imported in SpaceClaim.

Fig. 5.22. Geometrical model of the fender used to obtain the mathematical model (using Finite

Element Method)

34

5.3.2. Mathematical model with finite elements of the fender The geometrical model of the fender was reshaped in order to reduce the geometrical

configurantion of the virtual 3D model. The geometric model is characterized by 322 geometrical points,520 segment of boundary and 199 surfaces (fig. 5.24 a).

Fig. 5.24. a) 3D model imported in ANSYS; b) Model with finite elements ( different angles of

view).

5.3.3. Material properties. Boundary conditions Boundary conditions The fender wasconsidered fixed in the 7 points as shown (same way the experimental

measurements were done). For each fixed point there was considered a local Carthesian coordonate system XOY parallel with the fixing points, and axis OZ being perpendicular on the fixed surface (fig. 5.26 – 5.30). In this way in the node points within 8 mm distance from the center of coordinate system there have been imposed displacements and rotation with null values.

For the armonic analysis (fig. 5.31), the perturbation force with constant amplitude (10 N) was applied in the middle of the fender, on the interior margin (above the wheel), place that is similar to the one from experimental measurements in which the load was applied.

Fig. 5.26. Boundary conidtions for modal and armoinic analysis. Location and fixed points for

P1-P7 (different angles of view).

35

5.3.4. Results For all the 7 models, there has been determinde (and shown) frequencies and own

vibration modes under the form of maps with colours and graphs. In tabel 5.5 there has been highlited the comparison (under the form of percentage errors)

between the experimental values and the computed ones. Tabel 5.5. Frequency values measured experimentaly and computed with ANSYS.

Material Value Frequency [Hz]

1 2 3 4 5 6 7 8 9 10

Steel

Test 49.96 149.92 166.19 212.80 247.53 279.15 321.97 358.00 435.44 486.66 ANSYS 48.28 149.36 167.08 216.98 246.03 268.46 321.61 338.84 427.78 479.71 Error [ %] -3.37 -0.37 0.54 1.97 -0.61 -3.83 -0.11 -5.35 -1.76 -1.43

Kevlar

Test 50.63 94.21 105.85 136.60 265.98 ANSYS 50.05 84.81 113.20 129.10 266.88 Error [ %] -1.14 -9.97 6.95 -5.49 0.34

Hemp

Test 50.02 119.80 149.96 153.89 155.59 221.06 387.52 ANSYS 51.40 118.88 143.18 152.20 158.79 220.10 375.64 Error [ %] 2.75 -0.77 -4.52 -1.10 2.06 -0.44 -3.07

Cotton

Test 34.01 49.98 92.39 98.15 215.56 239.18 342.56 373.09 ANSYS 34.26 50.05 94.80 100.19 219.54 247.11 346.27 377.54 Error [ %] 0.75 0.14 2.61 2.08 1.85 3.31 1.08 1.19

Glass fiber

Test 41.20 110.53 191.74 228.67 253.15 ANSYS 42.07 112.12 194.44 218.91 261.38 Error [ %] 2.12 1.44 1.41 -4.27 3.25

Carbon-kevlar

Test 39.76 49.96 87.37 103.55 143.30 149.90 161.20 189.78 214.66 ANSYS 42.96 52.63 88.01 106.22 147.46 159.30 164.58 201.93 211.84 Error [ %] 8.04 5.34 0.73 2.57 2.90 6.27 2.10 6.40 -1.31

Carbon

Test 50.03 70.27 143.77 148.23 ANSYS 50.01 72.09 143.68 152.40 Error [ %] -0.04 2.59 -0.07 2.81

Results on the fender made of steel slab In the domain of studied frequencies (10-500 Hz), there have been identified 11

frequencies, the own modes of vibrations being shown in fig. 5.32 -5.42. Variation of axial displacement function of frequency, computed in the control points (middle and side), with the respresentation of complex values (real and imaginar) and amplitudes are shown in fig. 5.43-5.45.

Fig. 5.32. Own vibration mode corresponding Fig. 5.42. Own vibration mode corresponding to frequency of 48,276 Hz (steel). to a frequency of 479,71 Hz (steel).

36

Fig. 5.45. Frequency response (material: steel slab). Displacement amplitude [mm] axial (axis

OY), measured in both points.

5.4 Study of the structural stability of the fender by fluid-structure analysis

The structural stability of the fender was done in two phases: - In the first phase there was realized a study of the air flow around the fender,

considering that the air is perpendicular on the fender (laterla wind) at a speed of 100 km /h. After the computations done in the fluid domain, was determined the relative pressure distribution that pushes on the exterior part of the fender.

- In the second phase the relative pressures were considered to be structural loads, determined by static analysis, the distribution of displacements and equivalent tensions (Von Mises) in the fender as well as the stability factor of the structure that was determined by specific stability analysis.

5.4.1. Analysis of fluid flow around the fender Generating the model with finite elements It was obtained a finite volume charcterized by 432760 tetrahedral cellsand 1503647

ponts (nodes), having a finer mesh around the fender (fig. 5.46 b, c and d). The limit layer around the fender and og the side wall is represented in fig. 5.46 b), c).

Fig. 5.46. Model with finite elements. a) air domain around the fender; b) model with finite

elements; c) model with finite elements (shown in section); d) detailed model with finite volumes around the fender.

37

Boundary conditions. Solving the system In order to apply boundary conditions 3 regions have been generated (fig. 5.47 a, b):

- The region were the air enters the fluid domain,named region of type „Inlet”, considered at an air spede of 100 km/h, that acts over axis OY, perpendicular on transversal area (Vx=Vz=0), having an average turbulance of 5%;

- Region were the air exits the fluid domain called „Outlet”, from which the air exits without oposing resistance (null pressure drop ΔP=0);

- Wall region, in which speeds are null (Vx=Vy=Vz=0), region made of lateral surface of domain and of surfaces around the fender.

Results After finding the solution it has been determined that the maximum value of the pressures

on the fender surface (629,36 Pa, fig. 5.48 a ), as well as its distribution. The flow speed around the fender is represented in fig. 5.48 b, as the form of flowing lines. The distribution of relative pressure determined by calculus was transfered on the structural model as a load.

Fig. 5.47. a) Regions used to apply the boundary conditions; b) Boundary conditions applied; c)

Convergence graph of the process variables

Fig. 5.48. a) Relative pressure distribution [Pa] at the level of exterior surfaces of the fender;

b) Speed field [m/s] determined for flow simulation.

5.4.2. Stability calculus Stability calculus was performed on all the 7 fender models presented in Ch 5.3. For static analysis,performed in order to determine the equivalent tensions distribution

(Von Mises) and of axial displacement under the action of relative pressures (determined by CFD computations), same finite element models were used (fig. 5.23 – 5.25) as well as support conditions (fig. 5.26 – 5.30) presented in Ch 5.3. Moreover as a static load it was considered the relative pressure distribution (due to an air flow of 100 km/h), transfered in ANSYS structural from ANSYS CFX (fig. 5.49).

38

Fig. 5.49. Structural load. Distribution of relative pressure [Pa] at the level of exterior surfaces of

the fender imported in ANSYS. Maximum values of axial displacements (axa OY), Von Misses tensions and stability

factors are summarized for al the 7 fender models in tabel5.6. Tabel 5.6. Results obtained for the 7 fender models (stability analysis).

Nr. Fender material

Obtained results

Figure nr. Static analysis Analysis at

stability Axial

displacement [mm]

Von Mises tensions [MPa]

Stability factor

1 Steel 0.31 34.89 15.68 5.50 2 Kevlar 0.89 12.06 8.28 5.51

3 Hemp/ epoxy resin 0.64 3.95 21.60 5.52

4 Cotton/epoxy resin 2.43 9.47 3.62 5.53

5 Glass fiber 0.83 9.04 10.20 5.54 6 Carbon-Kevlar 1.03 16.65 5.05 5.55 7 Carbon 0.64 14.15 10.28 5.56

Fig. 5.50. Results fender - material: steel. a ) Von Mises tension distribution [MPa] (different

angles of view); b) axial displacement distribution [mm], axis OY; c) minimum multipling factor of the load and the mode of structural stability loss.

39

5.5. Conclusions 1. By analyzing the values from tabel 5.5, the values of own frequnecies determined by

Finite Element Method are in compliance with the ones determined by experiments, the maximum error being of 9,97% between these values being:

o 5,35% for the fender made of steel: o 9,97% for the fender made of kevlar; o 4,52% for the fender made of hemp; o 3,31% for the fender made of catton; o 4,27% for the fender made of glass fiber; o 8,04% for the fender made of carbon-kevlar; o 2,81% for the fender made of carbon.

2. The greatest difference of all the measured frequencies are the one for the kevlar

fender.

3. the geometrical model of the fender, also the model made with finite elements are in compliance with the physical ones and cand be used for future studies.

4. From the static analysis, it cand be observed that the values obtained for the tensions in the fender structure subjected to pressure, are in the elastic zone and are much more lower than the yielding stress of the material for all the 7 materials studied. Therefore the material is subjected to loads only in the elastic domain, therefore Hooke’s law is valid.

5. For the fender made of steel sheet, the static analysis shows the smallest axial displacment (0,31 [mm]), the largest values of the axial displacement being for the one made of cotton (2,43 [mm]) (tabel 5.6). This shows that the studied fenders, without the hemp one, have to be with bigger thicknesses than the one presented in order to have displacements similar to the fender mate so steel sheet. The displacement for the cotton fender is explicable dur to the fact that the material used has a low value for the elasticity modulus.

6. The maximum stability factor is obtained for the fender made from hemp and is with 37,75% bigger than the stability factor of the steel fender (tabel 5.6). This values is due to grater thickness ( 2,4 [mm]).

7. The smallest stability factor was obtained for the cotton fender and is 4,33 times smaller than the stability factor of the steel fender (tabel 5.6), but big enough in order to satisfy the conditions of the stability analysis presented before. This value is due to the thickness (1,2 [mm]) as well to the elasticity modulus (6000 [MPa]) used for this fender.

8. Neglecting the hemp fender, the carbon and glass fiber fenders have a stability factor close to the one made fromm steel sheet.

9. The study shows that composite materials based on carbon and fiber glass can be used in car industry as body parts.

10. The study shows that for static analysis, liniar ones as the modal is, armonic analysis of structural stability, composite materials cand be represented by equivalent properties, isotropic and homogenous..

40

Ch.6. Capitalization of results, contributions and future research

6.1. Capitalization of the researched results Theoretical studies, numerical caculus and the results of the experimental research

presented in this thesis have been published in articles rated ISI (6 articles) , with an impact factor, in speciality magazines that were indexed and presented at international confereneces (6 articles) as follows:

6.2. Original contributions I have determined a formula for the longitudinal elastic modulus computations of composite multilayer bars. I have dome from an elastic point of view an analogy with binding in series or in parallel of Hooke type models. For each layer in which the elastic properties are constant, a Hooke model is associated and for which the elastic constant depends on the lenght of the sector and thickness from which it is made of. Hooke models attached to every layer that has constant properties are considerd to be binded in parallel, determinig an equivalent model, and these model binded in a series give the global Hooke model having thus the elastic modulus of the bar. I have realised sets of test bars made from epoxy resin reinforced with fibers from glass,carbon,kevlar,carbon-kevlar. As well I have done sets of test bars made from epoxy resin reinforced with hemp and cotton. Ununiformities have been realized by changing the number of interrupted layers and by their leghts. All test bars were subjected to traction tests. I have determined the characteristic curves, Young modulus, failiure resistance and I have computed three factors that give ununiformity. For the study ou ununiformities and for comparison with the clasical composite materials, I have realized sets of test bars from natural dammar resin reinforced with hemp,cotton,mix of cotton and in, mix of silk and cotton. Also for these test bars I have determined the characteristic curves ,Young modulus,failire elongation, failiure resistance, elasticity factors,resistance factors and uniformity factors. For the study of transversal vibrations of bars with rectangular cross sections I have proposed a new field of displacements variabil on the bars width. I have introduced new relations for computing the inertial parameters and rigidity for the section of the bars, for which the movement equation for transversal vibrations is formally identical to the one in Euler-Bernolli theory. I have generalized the movement equation taking into account three damping mechanisms found in the speciality literature. I have studied free vibrations of the simple supported bar using the transformed Fourier in sinus. Experimental studies have been performed on four sets of composite materials having as matrix a combination with 75% naturala dammar resin and 25% epoxy resin reinforcement being done with in,cotton,silk and hemp. For multilayer composite bars having symmetry with respect to the median plane, I proposed a state of nonlinear displacements. I have introduced new inertial characteristics ( Aρ , ,1Iρ 2Iρ ) and also rigidity ones ( GA , 1EI , 2EI ) for the cross section of the bar and I have particularized these characteristics for homogenous isotropic bars I have worked on movement equations giving them a form similar to the clasical Timoshenko ones. If in the clasical bar theory of Timoshenko is used only one ununiformity coefficient for tensions, in the presented model there are three such coefficients that characterize the ununiformity tensions in the cross section of the bars.One coefficient takes into account the tangential tension variations, as in clasical theory, and the other two take into account the normal tensions. I have particularized these coefficients for homogenous and isotropic bars and have noticed that in this case the movement equations are the same as the Timoshenko ones from

41

clasical theory. Moreover by the customization of these coefficients mathematical models can be obtained for laminated composite bars, also found in the speciality literature. For a sandwich bar I have determined the calculus relations for inertial characteristics and also for rigidity and for ununiformity coeffcients function of the ratio between the thcikness of the core and thickness of the bar and also for the mass and elastic properties of the core and of the exterior layers. I have computed the inertial characteristics,rigidity,ununiformity coefficients for numenrical values of a experimental bar. With these results I have computed the own pulsation for the bar. Comparing the result obtained by me with the one obtained using the Euler-Bernoulli simplified theorem I have noticed that the difference in values are very small. Thus the Euler-Bernoulli theorem can be used for sandwich bars I have realized sandwich bars with the core made from honeycomb polypropylene with thickness of 10 mm, 15 mm and 20 mm. At the exterior there were one or two layers of glass fiber and carbon fiber. I have determined experimentaly the frequncy and the damping factor for the bars made of the mentioned materials having widths of 30 mm, 45 mm, 60 mm having a fixed end and a free end with variable leght between 200 mm and 350 mm. I have made a physical car body part (left fender of a Peugeot 206 made in 1998 ) from composite materials based on epoxy resin reinforced with fibers of:carbon,kevlar,carbon-kevlar,glass,cotton and hemp. For different such fenders I have realized an experimental modal analysis so to find the own vibrations with respect to a certain direction, an armonic analysis and a Finite Element one and also a study regarding the structural stability of the fender by fluid-structure interaction. I have done measurements in two points of the fender , for each determining the response function in frequecny and own vibration modes. I have obtained the 3D model of the fender, used in the virtual analysis model and for experimental study with the Finite Elemenet Method, making a scan of the actual fender and obtaining the virtual mesh. I have determined the own modes of the fender using the modal extraction method of Block-Lanczos. The modal analysis were preceded by armonic analysis,considering a disturbing force of constant amplitude over the frequency domain studied.

I did a static analysis and determined the values of the parameters used in the stability analysis, in order to find the multiplication factor of the load from which stability is lost. I made the stability analysis and studied the way in which the fender loses its stability and determined the minimum multiplication factor for which the fender loses its stability.

6.3. Future research paths

The following research paths can be followed in the future: - the development of composite materials based on natural resins such as the Manila tree, mastic,shellac etc - development of new reinforcing materials that are very often encounterd such as: cattail,reed,straws,rye - obtaining some materials that are composite hybrid in which a mix of natural and synthetic fibers is made - study of other properties for the composite materials like phonic isolators and thermal conductivity. - manufacturing of other car parts . - study of interlaminar phenomenon that appear when vibration occur for composite multilayer bars.

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Bibliografie (selectivă) [ABO_2017]-Abo Sabah, S.H., Kueh, A.B.H., Al-Fasih, M.Y., (2017), Comparative low velocity impact behavior of bio-inspired and conventional sandwich composite beams, Science and Technology, 149, 64-74. [ASS_2015]- Mustapha Assarar, Wajdi Zouari, Hamid Sabhi, Rezak Ayad, Jean-Marie Berthelot (2015), Evaluation of the damping of hybrid carbon-flax reinforced composites, Composite Structures, 132, 148-154. [BOL_2017(1)]-Bolcu A., Dumitru N., Ciucă I., Stănescu M., Bolcu D., (2017), The vibrations study of Dammar based composite bars reinforced with natural fibers by using a new Euler-Bernoulli theory, Mater. Plast., 54(3), 423-429. [BOL_2017(2)]-Bolcu D., Stănescu M.M., Ciucă I., Miritoiu C., Bolcu A., Ciocoiu R., (2017), The vibrations study of Dammar based composite bars by using a new Euler-Bernoulli theory, Mater. Plast., 54(1), 1-7. [CHE_2017]-Chenjun Liu, Zhang, Y.X., Jing Li, (2017), Impact responses of sandwich panels with fibre metal laminate skins and aluminium foam core, Composites Structures, !82, 183-190. [CIU_2017]-Ciucă, I., Bolcu, A.,Stănescu, M.M., (2017), A study on mechanical properties of bio-composite materials with a dammar based-matrix, Enviromental Engineering and Management Journal, 16(12), 2851-2856. [DAO_2017]- Hajer Daoud, Abderrahim El Mahi, Jean-Luc Rebiere, Mohamed Taktak, Mohamed Haddar (2017), Characterization of the vibrational behavior of flax fibre reinforced composites with an interleaved natural viscoelastic layer, Applied Acoustics, 128, 23-31. [GUE_2016(1)]- Marie Joo Le Guen, Roger H. Newman, Alan Fernyhough, Grant W. Emms, Mark P. Staiger (2016), The damping-modulus relationship in flax-carbon fibre hybrid composites, Composites Part B: Engineering, 89, 27-33. [HAJ_2018]- Hamzeh Hajiloo, Mark F. Green, John Gales (2018), Mechanical properties of GFRP reinforcing bars at high temperatures, Construction and Building Materials, 162, 142-154. [HER_2008]-Hermann L., (2008), Vibration of the Euler-Bernoulli Beam with Allowance for Dampings, Proceedings of the World Congress on Engineering 2008 Vol II ,WCE 2008, July 2 - 4, London, U.K. [HUA_2017]-Huang, S.Y., Lou, C.W., Yan, R., Lin, Q., Li, T.T., Chen, Y.S., Lin, J.H., (2017), Investigation on structure and impact-resistance property of polyurethane foam filled three-dimensional fabric reinforced sandwich flexible composites, Composites Part B: Engineering, 131, 44-49. [JON_1999]- Jones, R. M., Mechanics of Composite Materials Second Edition, ISBN: 1-56032-712-X Philadelphia, 1999. [KHO_2017]- S.M. Khoshnava, R. Rostami, M. Ismail, A.R. Rahmat, B.E. Ogunbode (2017), Woven hybrid Biocomposite: Mechanical Properties of woven kenaf bas fibre/oil palm empty fruit bunches hybrid reinforced poly hydroxybutyrate biocomposite as non-structural building materials, Construction and Building Materials, 154, 155-166. [KUL_2018]- P. Kulkarni, A. Bhattacharjee, B. K. Nanda (2018), Study of damping in composite beams, Materialstoday: PROCEEDINGS, 5, 7061-7067. [KUM_2017]- N. Kumar, D. Das (2017), Fibrous biocomposites for nettle (Girardinia diversifolia) and poly(lactic acid) fibers for automotive dashboard panel applications, Composite Part B: Engineering, 130, 54-63. [MAJ_2009]-Majkut, L., Free and forced vibrations of Timoshenko beams described by single difference equation, (2009), Journal of Theoretical and applied mechanics, Warsaw, 47 (1), 193-210. [OQL_2014]- Faris M. Al-Oqla, S.M. Sapuan (2014), Natural fiber reinforced polymer composites in industrial applications: feasibility of date palm fibers for sustainable automotive industry, Journal of Cleaner Production, 66, 347-354.

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[QIN_2009]-Qin, Q.H., Wang, T.J., (2009), An analytical solution for the large deflection of a slender sandwich beams with a metalic foam core under traverse loadind by a flat punch, Composite Structures, 88, 509-518. [QIN_2017]-Qin, Q., Xiang, C., Zhang, J., Yuan, C., Wang, M., Wang, T.J., Poh, L.H., (2017), On low-velocity impact response of metal foam core sandwich beam: A dual beam model, Composite Structures, 170, 1039-1049. [QIN_2018]-Qin, Q., Zheng, X., Zhang, J., Yuan, C., Wang, T.J., (2018), Dynamic response of square sandwich plates with a metal foam core subjected to low-velocity impact, International Journal of Impact Engineering, 111, 222-235. [SAT_2016]-Sathishkumar, T.P., Naveen, J., Navaneethakrishnan, P., Satheeshkumar, S., Rajini, N., (2016), Characterization of sisal/cotton fibre woven mat reinforced polymer hybrid composites, Journal of Industrial Textiles, 47, 429-452. [SHE_2018]- Shamim A. Sheikh, Zahara Kharal (2018), Replacement of steel with GFRP for sustainable reinforced concrete, Construction and Building Materials, 160, 767-774. [SWO_2014]- Y. Swolfs, L. Gorbatikh, I. Verpoest (2014), Fibre hybridization in polymer composites: A review, Composites Part A: Applied Science and Manufacturing, vol. 67, p. 181-200. [VIK_2015]-Vikas Sahu, Keshav Sings Bisen, Murali Krishna, (2015), Mechanical properties of sisal and pineapple fiber hybrid composites reinforced with epoxi resin, International Journal of Modern Engineering Research, 5, 32-38. [WAL_2017]-Walsh, J., Kim, H.I., Suhr, J., (2017), Low velocity impact resistance and energy absortion of environmentally friendly expanded cork core-carbon fiber sandwich composites, Composites Part A; Apllied Science and Manufacturing, 101, 290-296. [WAN_2017]-Wang, T., Qin, Q., Wang, M., Yu, W., Wang, J., Zhang, J., Wang, T.J., (2017), Blast response of geometrically asymmetric metal honeycomb sandwich plates: Experimental and theoretical investigations, International Journal of Impact Engineering, 105, 24-38. [WIS_2016]- M.R. Wisnom, G. Czel, Y. Swolfs, M. Jalavand, L. Gorbatikh, I. Verpoest (2016), Hybrid effects in thin play carbon/glass unidirectional laminates: Accurate experimental determination and prediction, Composites Part A: Applied Science and Manufacturing, vol. 88, p. 131-139. [YUS_2016]-Yusoff, R.B., Takagi, H., Nakagaito, A.N., (2016), Tensile and flexural properties of polylactic acid-based hybrid green composites reinforced by kenaf, bamboo and coir fibers, Industrial Crops and Products, 94, 562-573. [ZHA_2016]-Zhang, J., Qiu, Q., Xiang, C., Wang, T.J., (2016), Dynamic response of slender multilayer beams with metal foam cores subjected to low-velocity impact, Composite Structures, 153, 614-623. [ZUC_2018]- B. Zuccarello, G. Marannano, A. Mancino (2018), Optimal manufacturing and mechanical characterization of high performance composites reinforced by sisal fibers, Composite Structures, 194, 575-583. [WWW_5]. http://compositeslab.com/where-are-composites-used/automotive-applications/ [WWW_7]. http://www.3mb.asia/carbon-fiber-usage-in-automotive-industry/ [WWW_14].http://www.imst.pub.ro/Upload/Sesiune/ComunicariStiintifice/Lucrari_2015/06.15/15_L08.pdf

http://compositeslab.com/where-are-composites-used/automotive-applications/

http://www.3mb.asia/carbon-fiber-usage-in-automotive-industry/

http://www.imst.pub.ro/Upload/Sesiune/ComunicariStiintifice/Lucrari_2015/06.15/15_L08.pdf

http://www.imst.pub.ro/Upload/Sesiune/ComunicariStiintifice/Lucrari_2015/06.15/15_L08.pdf

Abstract PhD THESIS - Universitatea din Craiova · 2019-07-09 · Abstract . PhD THESIS ....

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