CC BY
64
УДК 624.04
R.A. Turusov, H. Memarianfard
MGSU
NUMERICAL PREDICTION OF RESIDUAL STRESSES IN OPEN-ENDED THICK-WALLED CROSS-PLY FILAMENT WOUND FIBER-REINFORCED CYLINDERS
In this paper a three-dimensional finite element analysis employed to predict thermal residual stresses field which arises during the cooling stage at the free edges of a thick walled filament wound cylinder with cross-ply lamination. The inner radius of composite is 50 mm and outer radius is 75 mm and the thickness of steel mandrel is 3 mm. The results showed that the radial stresses near the free ends of the cylinder increased two times compared to radial stresses in the middle of the cylinder and interlaminar shear stresses exceeded 6 MPa close to the free edges.Thus, a two-dimensional stress analysis does not fully reflect the complex state of stress of thick-walled cross-ply filament wound cylinders.
Key words: cross-ply, filament wound, interlaminar stresses, free edge, thick-walled, hoopcracks, cooling
Introduction
Composites cylinders and tubes are widely used for a range of industrial applications and have been the subject of many previous studies. The manufacture of composite tubes by filament winding generates process-induced stresses [1]. In the production process of a thick-walled filament wound fiber reinforced cylinder due to anisotropic thermal shrinkage during cure and cool down process after curing, residual thermal stresses arise. It can also lead to delamination and interlaminar cracks as shown in Fig. 1. Experiments over the thick-walled unidirectional (90) and cross-ply (90/0) filament wound cylinder with open ends show that for unidirectional winding, thick-walled cylinder can be obtained monolithic and without cracks, but for cross-ply (90/0) winding, the cracks and delamination were observed in the cylinder.
Many analytical and numerical models have been developed for modeling and prediction the thermal residual stresses generated in thick-walled fiber reinforcement filament wound cylinder during cure and cooling processes. R.A. Turusov, V.A. Ko-rotkov and B.A. Ro-senberg in 1981 considered viscoelastic thermal stresses arising in the thick-walled unidirectional filament wound cylinder of composite material in the process of cooling and storing, and taking into account the different speeds of cooling [2, 3]. R.A. Turusov, V.A. Korotkov and B.A. Rosenberg predicted technological thermal stresses during cure and cool-down
Fig. 1. Hoop cracks in thick-walled filament wound cylinder after curing and cooling processes
in thick-walled filament wound unidirectional cylinder and considered transient heat conduction with viscoelastic stresses analysis [4]. B.C. Ekelchik and his colleagues examined the thermal stresses in orthotopic thick-walled cylinders of fiber-reinforced polymer under non-uniform cooling [5]. М.W. Hyer, David E. Coopera & David Cohen determine stresses and deformations in cross-ply composite tubes subjected to a uniform temperature change [6]. But most of the previous works on predication of thermal residual stresses in thick-walled filament wound cylinders are limited and focused on two dimensional analyses with a planar elasticity solution, and three-dimensional stress and strain reduced to two-dimensional plane stress or strain state. The results of residual thermal stresses from the two-dimensional plane stress or plane strain model of unidirectional and cross-ply filament wound cylinders when composite is simplified to a homogenous orthotropic material show that radial stresses in cross-ply filament winding are smaller than the radial stresses in unidirectional filament wound cylinder, so two-dimensional analyses of residual stresses can't explain the reason of delamination of cross-ply filament wound cylinder.
For laminated composites, it is well-known that at the free edges interlaminar stresses arise from the mismatch of elastic properties between layers [7, 8]. The first analytical method dealing with anisotropic materials was proposed by Puppo and Evensen [9] to evaluate interlaminar shear of anisotropic layers separated by isotro-pic shear layers with interlaminar normal stress being neglected throughout the laminate. The first numerical studies of interlaminar stresses under mechanical loading were presented by Pipes and Pagan using finite difference method [10], Rybicki, 1971 [11], Wang and Crossman, 1976 [12], Murthy and Chamis, 1989 [13] considering interlaminar stresses using finite element method but the majority of these works focused on interlaminar stresses under an uniaxial loading or uniform bending of laminated composites. For cross-ply filament wound cylinder with open ends due to the difference between the CTE (coefficient of thermal expansion) of longitudinal (0°) and transversal (90°) layers along the axial direction (for cross-ply glass fiber reinforced composite K = a z 90/a z 0 «3) arises a mismatch axial strain Asz due to change of temperature in the cylinder and it can lead to interlaminar shear stresses as shown on the Fig. 3. Thus in these regions near free-edge and ply cracking, it has been recognized that the stress state is three-dimensional in nature and not predictable accurately by the classical lamination theory [14].
Fig. 2. Free edge effect of cross-ply (0/90) laminated under tension [15]
Fig. 3. Mismatch of axial strain of cross-ply filament wound cylinder with open ends
Most Polymer matrix fiber reinforced composites are usually cured at an elevated temperature up to 150...350 °C and then cooled to ambient conditions. The residual stresses arising in the cure process of polymer composites are very small and the main part of the residual stresses arise during cooling process [15]. In the present paper, we consider uniform temperature change of cylinder from 110 to 20 °C and linear elastic behavior of the material. We employed three-dimensional finite element formulation to determine stress-strain state in the cylinder.
Finite element formulation
Three dimensional elastic constitutive relation (stress-strain relationship) with strains caused by temperature changes and strains caused by applied loads can be expressed as:
sij = Cjki 8h = d aa
(e kl 8 Id);
(1)
Where four rank stiffness tensor C u has 81 components but due to symmetry they are reduced to 36 components for fully anisotropic materials. To simplify the problem we assume that physical and mechanical properties of material are constant and doesnot varies with temperature in cooling process. The linear stress strain relation for cross-ply orthotopic fiber reinforced composite can be represented as follows [16]:
(2)
C =
13
s1 C11 C 12 C 13 0 0 0 81 — a1DT
S2 C 21 C 22 C 23 0 0 0 82 — a2 ДТ
S3 C 31 C 32 C 33 0 0 0 83 — a3DT
T12 0 0 0 C 0 0 812
T23 0 0 0 0 C55 0 8 23
T31 _ 0 0 0 0 0 C66 _ _ 831
C11 = 1 — V23V32 E2 E3 Д C = 1 22 E1E3 Д ' C 33 = 1 — V12V21 E1E2 Д '
V31 - "V21V32 C = 23 V32 — V12V31 C44 = C55 = ^23. C
E2 E3 д E1E3D 66
C = V21 - V23V31
E2 E3 Д
1 —vv —vv —vv — 2 v v v
д _ 12 21 23 32 v31v13 21 32 13
E1E2 E3
13
For static problems, the equilibrium equations can be written as: LT s + fb =0, (3)
where fb is the external force and the three-dimensional strain operator L is given by [17]:
LT =
д 0 0 д 0 д
ÔXj дх2 дх3
0 д 0 д д 0
дх2 дх1 дх3
0 0 д 0 д д
дх3 дх2 дх1
(4)
Using the Galerkin method, we can multiply the Equation (3) by a virtual displacement field Su and integrate over the domain V currently occupied by the body [18]:
J( LT s + fb y>uTdV = 0.
(5)
The Equation (5) can be satisfied if the quantity in the parentheses vanishes. Hence using divergence theorem we obtain from (5) [19]:
J(L5u )T sdV = J fb duTdV + j5uTtdS. (6)
V V S
With the boundary conditions X n = t or u = up are prescribed on complementary parts of the body surface S, where all integrals extend over the element domain Ve of each of the n elements of the finite element mesh. The displacements u, v, w within an element are interpolated from nodal degree of freedom d and in terms of shape functions N. is:
u = X Nu, v = X Nv, w = X Nw,. (7)
The stress and strain-displacement equations are expressed in matrix form as:
[B ] = [L ][N ]; (8)
{} = [B ]{u}; (9)
{s} = [C ][ B ]{u}. (10)
With the help of the Equations (6)—(10) where all integrals extend over the element domain V of each of the n elements of the finite element mesh, the weak form of the balance of momentum can be reformulated as:
X J[ B ]T sdV = X JV [ N ]T fdV + X J[ N ftdS.
The external force vector:
fet = XJ[ N ifdV + XJ[ N ftdS•
n V n V
And the internal force vector: fml =XJ[ B ]T sdV •
(11)
(12)
(13)
The general form of the Equation (11) can be presented in the following form:
[ * ]{u} = { f};
{u} = [ K ]-1 {f}, where [K] in the Equation (14):
K = XJ[5r [C][5] dF.
n y
The model is meshed with 20-nodes brick elements as shown in Fig. 4. The 20-node hexahedron is the analog of the 8-node "serendipity" quadrilateral. The 8 corner nodes are augmented with 12 side nodes which are usually located at the midpoints of the sides.
Using polynomial functions and natural coordinate systems (Z, h, Z) and -1 < Z, h, Z - 1. The shape functions of the 20-node hexahedron can be grouped as follows. For the corner nodes [20]:
1
(14)
(15)
Fig. 4. The 20-node hexahedron (serendipity) element
N = 8 (+ & X1)(1+ ZZ )( + + & - 2);
N = 1)(( + nn.)(( + ZZ.X . = 2, 6, 14, 18; N. = 4((-П2)(( + &)(( + ZZ.X i = 4, 8, 16, 20; N. = -4(-Z2)(l + &)(1 + ПП.X i = 9> 10> 11. I2-
(16)
The matrix 5 = LN involves differentiation with respect to the global coordinates x. However, the shape functions are functions of the isoperimetric coordinate Z. For this reason the chain rule is used to obtain:
ON.
^z"
dN
dx dN. r ON, --'- = J-
5Z dx
1 dN
dx
(17)
= J
dx cC,
Eventually the internal force (13) can be presented in the following form: fnt =u:J" J'ldet J)[5] sdZdhdx.
(18)
Analyses:
In this paper a three dimensional Finite Element Analysis (FEA) is used to define three-dimensional (3D) stress state in thick-walled filament wound cross-ply laminated cylinder. The ANSYS finite element code is used for the present analysis.
The outer radius of the mandrel r = 50 mm and the thickness of mandrel is 3 mm. The outer radius of the filament wounded cylinder is R = 75 mm and the length of the cylinder L = 200 mm. A cross-ply laminate layup of (90/02/902/02/902/02/902/02/ 90 /0 /90) used for present model as shown in Fig. 5, 6.
90°
0°
0°
90°
90°
0°
0°
90°
Fig. 5. Example of cross-ply lay-up
Fig. 6. Unidirectional fiber-reinforced laminated a, Cross-ply fiber-reinforced laminated b
The results of three-dimensional stress analysis of thick-walled filament wound cross-ply laminated cylinder with free ends are given with respect that displacements and stresses are dependent not only on radius r but also on length Z-axis (Fig. 7).
Fig. 7. The scheme of thick-walled laminated composite cylinder with free ends Distribution of residual radial stresses c and shear stresses t over the thickness
r rz
(in the middle and ends of the cylinder) and over the length of cylinder after cooling to room temperature is shown on Fig. 8—11.
Thickness of cylinder, mm
Fig. 8. Distribution of radial stresses cr over the thickness of filament wound cross-ply cylinder (Solid line: near to the free edges; dash line: in the middle)
-100
-50
7 j 6 -5 -4 --
3 -2 -1 --
2 Br
50
100
Z-axis, mm
Fig. 9. Distribution of maximum radial stresses over the length of cylinder (Z-axis)
Thickness of cylinder, mm
Fig. 10. Distribution of shear stresses trz over the thickness of filament wound cross-ply cylinder close to the free edges
7 г
г 5..
3 ■■
1 ■■
-100
-во
-60
-40
20
40
100
-20 -1 -3
-5 + -7 -L
Z-axis, mm
Fig. 11. Distribution of shear stresses x^ over the length of the cylinder
Conclusions:
From the curves of Fig. 8 we can see that radial stress increases in transversal (0°) layer and decreases in longitudinal (90°) layer and they vary sharply across the free edges due to the differences of physical properties in each layer. The results of three-dimensional finite element analysis of thick-walled cross-ply filament cylinder with free ends show that radial stress two times increased over the free edges and these might lead to the occurrence of delamination of the laminate layers and thus lead to the failure of the entire structure. Fig. 11 shows the shear stress distribution xrz over the length of the cylinder. It can be seen that shear stress is equal to zero in in the middle of the cylinder but shear stress increased rapidly along the free edges to 6.2 MPa. So delamination and tangential cracks can arise at the free edges of the cylinder and spread throughout the length of the cylinder. This interlaminar stresses near the edges of the cylinder arise due to a mismatch of axial strain Dez. Thus the smaller differences between az of the layers can lead to smaller shear stresses and edge effects in laminated cylinders with open ends. The results of the three-dimensional stresses analyses finally show that two-dimensional (plane-stress or plane-strain state) analysis of stresses can't accurately predict the residual thermal stresses in thick-walled filament wound cylinders.
References
1. Casari P., Jacquemin F., Davies P. Characterization of Residual Stresses in Wound Composite Tubes. Applied Science and Manufacturing. February 2006, vol. 37, no. 2, pp. 337—343. DOI: http://dx.doi.org/10.1016/j.compositesa.2005.03.026.
2. Korotkov V.N., Turusov R.A., Andreevska G.D., Rosenberg B.A. Temperature Stresses in polymers and composites. Mechanics of composites. NY, March 1981, pp. 290—295.
3. Korotkov V.N., Turnsov R.A., Rozenberg B.A. Thermal Stresses in Cylinders Made of Composite Material During Cooling and Storing. Mechanics of Composite Materials. March 1983, vol. 19, no. 2, pp. 218—222. DOI: http://dx.doi.org/10.1007/BF00604228.
4. Korotkov VN., Turusov R.A., Dzhavadyan E.A., Rozenberg B.A. Production Stresses During the Solidification of Cylindrical Articles Formed from Polymer Composite Materials. Mechanics of Composite Materials. January 1986, vol. 22, no. 1, pp. 99—103.
5. Afanas'ev Yu.A., Ekel'chik V.S., Kostritskii S.N. Temperature Stresses in Thick-Walled Orthotropic Cylinders of Reinforced Polymeric Materials on Nonuniform Cooling. Mechanics of Composite Materials. July 1981, vol. 16, no. 4, pp. 451—457. DOI: http:// dx.doi.org/10.1007/BF00604863.
BECTHMK ü /2015
11/2015
6. Hyer M.W., Rousseau C.Q. Thermally Induced Stresses and Deformations in Angle-Ply Composite Tubes. Journal of Composite Materials. 1987, vol. 21, no. 5, pp. 454—480. DOI: http://dx.doi.org/10.1177/002199838702100504.
7. Roos R., Kress G., Barbezat M., Ermanni P. Enhanced Model For Interlaminar Normal Stress In Singly Curved Laminates. Composite Structures. 2007, vol. 80, no. 3, pp. 327—333. DOI: http://dx.doi.org/10.1016/j.compstruct.2006.05.022.
8. Liu K.S, Tsai S.W. A Progressive Quadratic Failure Criterion for a Laminate. Composites Science and Technology. 1998, vol. 58, no. 7, pp. 1023—1032. DOI: http:// dx.doi.org/10.1016/S0266-3538(96)00141-8.
9. Puppo A.H, Evensen H.A. Interlaminar Shear in Laminated Composite under Generalized Plane Stress. J Compos Mater. 1970, vol. 4, pp. 204—220.
10. Pipes R.B., Pagano N.J. Interlaminar Stresses in Composite Laminates under Uniform Axial Extension. Journal of Composite Materials. 1970, vol. 4, pp. 538—548.
11. Rybicki E.F. Approximate Three-Dimensional Solutions for Symmetric Laminates under In-Plane Loading. Journal of Composite Materials. 1971, vol. 5, no. 3, pp. 354—360. DOI: http://dx.doi.org/10.1177/002199837100500305.
12. Wang A.S.D., Crossman F.W. Calculation of Edge Stresses in Multi-Layered Laminates by Sub-Structuring. Journal of Composite Materials. April 1978, vol. 12, no. 1, pp. 76—83. DOI: http://dx.doi.org/10.1177/002199837801200106.
13. Murthy P.L.N., Chamis C.C. Free-Edge Delamination: Laminate Width and Loading Conditions Effects. Journal of Composites, Technology and Research. 1989, vol. 11 (1), pp. 15—22. DOI: http://dx.doi.org/10.1520/CTR10144J.
14. Dong S.B., Pister K.S., Taylor R.L. On the Theory of Laminated Anisotropic Shells and Plates. Journal of the Aerospace Sciences. 1962, vol. 29, no. 8, pp. 969—975.
15. Turusov R.A., Korotkov V.N., Rogozinskii A.K., Kuperman A.M., Sulyaeva Z.P., Garanin V.V, Rozenberg B.A. Technological Monolithic Character of Shells Formed from Polymeric Composition Materials. Mechanics of Composite Materials. 1988, vol. 23, no. 6, pp. 773—777. DOI: http://dx.doi.org/10.1007/BF00616802.
16. Autar K. Kaw. Mechanics of Composite Materials. Second Edition, CRC Press, November 2, 2005, p. 96.
17. Zienkiewicz O.C., Taylor R.L. The Finite Element Method for Solid and Structural Mechanics. Sixth Edition, Butterworth-Heinemann, 2005, p. 8.
18. Bathe Klaus-Jürgen. Finite Element Procedures. Prentice Hall, 1996, p. 171.
19. René De Borst, Mike A. Crisfield, Joris J.C. Remmers, Clemens V Verhoosel, Nonlinear Finite Element Analysis of Solids and Structures. Wiley, 2012, 540 p.
20. Zienkiewicz O.C., Taylor R.L. The Finite Element Method: Its Basis and Fundamentals. Sixth Edition, Butterworth-Heinemann, 2005, p. 121.
Received in September 2015.
About the authors: Turusov Robert Alekseevich — Doctor of Physical and Mathematical Sciences, Professor, Department of Strength of Materials, Moscow State University of Civil Engineering (National Research University) (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation; rob-turusov@yandex.ru;
Hamed Memaryanfard — postgraduate student, Department of Strength of Materials, Moscow State University of Civil Engineering (National Research University) (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation; hmemariyanfard@yahoo.com.
For citation: Turusov R.A., Memarianfard H. Numerical Prediction of Residual Stresses in Open-Ended Thick-Walled Cross-Ply Filament Wound Fiber-Reinforced Cylinders. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2015, no. 11, pp. 80—89.
Р.А. Турусов, Х. Мемарианфард
ЧИСЛЕННЫЙ ПРОГНОЗ ОСТАТОЧНЫХ НАПРЯЖЕНИЙ В ТОЛСТОСТЕННЫХ ДИАГОНАЛЬНЫХ НАМОТОЧНЫХ ЦИЛИНДРАХ ИЗ АРМИРОВАННЫХ ПОЛИМЕРОВ С ОТКРЫТЫМИ КОНЦАМИ
Трехмерный анализ методом конечных элементов использован для прогнозирования поля термических остаточных напряжений на свободных краях толстостенных намоточных цилиндров с диагональным расположением нитей в процессе охлаждения. Внутренний радиус композита — 50 мм, внешний радиус — 75 мм и толщина стального сердечника — 3 мм. Результаты показали, что радиальные напряжения вблизи свободных концов цилиндра выросли в два раза по сравнению с радиальными напряжениями в средине цилиндра. Межслойные напряжения сдвига превысили 6 МПа около свободных краев. Таким образом, двумерный анализ напряжений не отражает в полной мере сложное напряженное состояние толстостенных намоточных цилиндров с диагональным расположением нитей.
Ключевые слова: слой, поперечное расположение, армирующие волокна, метод намотки волокном, межслойные напряжения, свободный край, толстостенный, кольцевые трещины, охлаждение
Поступила в редакцию в сентябре 2015 г.
Об авторах: Турусов Роберт Алексеевич — доктор физико-математических наук, профессор кафедры сопротивления материалов, Национальный исследовательский Московский государственный строительный университет (НИУ МГСУ), 129337, г Москва, Ярославское шоссе, д. 26, rob-turusov@yandex.ru;
Мемарианфард Хамед — аспирант кафедры сопротивления материалов, Национальный исследовательский Московский государственный строительный университет (НИУ МГСУ), 129337, г Москва, Ярославское шоссе, д. 26, hmemariyanfard@ yahoo.com.
Для цитирования: Turusov R.A., Memarianfard H. Numerical Prediction of Residual Stresses in Open-Ended Thick-Walled Cross-Ply Filament Wound Fiber-Reinforced Cylinders // Вестник МГСУ. 2015. № 11. С. 80—89.
