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17Journal of Architectural and Engineering Research 2022-2 ISSN 2738-2656 DOI: https://doi.org/10.54338/27382656-2022.2-013
Hamlet Gurgen Shekyan1, Norayr Grigor Hovumyan2*, Aramayis Viktor Gevorgyan3
1 National Academy of Sciences ofArmenia, Yerevan, RA
2 National University of Architecture and Construction of Armenia, Yerevan, RA
3 Electromash GAM Scientific Production and Test Center (CJSC), Yerevan, RA
FUNDAMENTALS OF METHODOLOGICAL CONTROL OF THE STRENGTH OF COMPOSITE MATERIALS
The idea of this paper is to propose research methods designedfor strength analysis of composite materials which are widely used in load-carrying structures. It is shown that the strength analysis and study of composite materials are carried out in two directions - considering them as inhomogeneous composite materials, representing a regular multilayer medium of alternating reinforcement layers and a polymer binder, and the study of the issue of structural strength, in which composites are considered as homogeneous elastic orthotropic bodies, to which the theory of elasticity of anisotropic media is applicable. These approaches enables to represent the composites in the form of a continuous medium and use the methods of the theory of elasticity of anisotropic media in calculating the acting stresses, which makes it possible to use the influence of the elements of manufacturing technology on the physical and mechanical characteristics.
Keywords: composite materials (CM), strength, stiffness, elasticity
Introduction
Modern composite materials certainly now constitute one of the most important classes of engineering materials widely used in critical load-carrying structural elements. There are several reasons for this. One is that they offer highly attractive combinations of strength, stiffness, lightness and corrosion resistance. At present, the study of the strength of composite materials is carried out in two directions.
In the works where the first direction is applied [9, 10] composite materials are considered as heterogeneous composites that represent a regular multi-layer medium of alternating reinforcement layers and a polymer binder.
When applying this theory, certainly, difficulties arise in connection with the presence of defects in the manufacture of structures, etc.
In the works where the second direction is used [4, 8], the structural strength study of homogeneous elastic orthotropic bodies, to which the theoretical foundations of the elasticity of anisotropic media are applicable, is becoming widespread. This assumption is based on the fact that the dimensions of the reinforcing filler are negligibly small compared with geometric dimensions of the part's cross-section. Based on this, the composites can be represented as a continuous homogeneous medium. This approach makes it possible to use simple and well-developed methods of the theory of elasticity of anisotropic media when calculating the acting stresses, which enables to take into account the influence of elements of manufacturing technology on the physical and mechanical characteristics of a structure. Orthotropic composite materials are most widely used in load-carrying structures, therefore, the main attention is paid to this issue.
Materials and Methods
The strength characteristics of quasi-homogeneous, nonisotropic materials are derived from a generalized distortional work criterion. For unidirectional composites, the strength is governed by the axial, transverse, and shear strengths, and the angle of fiber orientation. The strength of a laminated composite consisting of unidirectional layers depends on the strength, thickness, and orientation of each constituent layer and the temperature at which the laminate is cured. In the process of lamination, thermal and mechanical interactions are induced which affect the residual stress and the subsequent stress distribution under external load. A
method of strength analysis of laminated composites is delineated using glass-epoxy composites as examples. The validity of the method is demonstrated by appropriate experiments.
Solution of the Problem
It is well known that in the theory of elasticity the stressed state of an anisotropic medium is described by the generalized Hooke's law
where aix are the components of the stress tensor; £im are strain tensor components; CUm are components of the elastic modulus tensor (which is a fourth rank tensor). In this case, the elastic potential is a function of the second degree, invariant with respect to the coordinate degree, then C = C . For an orthotropic medium, Equation (1) in expanded form can be represented as
ax =C11^x + Cn£y + C^
ay =C21^x + C22&y + C23^z az =C31^x + C32^y + C33^z ,
Tyz — C44vvz,
(2)
Tzx C55Vzx,
Txy — C66Vxy,
Hooke's law regarding the components of deformation will be
£x — an&x + au&y + auaz,
£y — a21ax + a22ay + a23az, £z — a31ax + a32ay + a33az , Vyz — a44^yz ,
v — a,
zx "-55 * zx ' Vxy — a66Txy,
where Cik are the elastic constants of the material, ^ are the constants of elastic deformation,
ak — Ck/ A ,
where A is the determinant, composed of the coefficients of the right-hand side of Equation (2), C^are the
corresponding minors of this determinant. In the expanded form, we will have [1, 9]
a11 — (C22C33 — C23 ) / A, a22 — (C11C33 — C13 ) / A, a33 — (C11C12 — C12 ) / A,
«12 — (C12C33 -C13C23)/ A; «13 = (C12C23-C |C 2)/ A, «23 =(C„C23-C £ 2)/ A,
a44 — C44 ; a55 — C55 ; a66 — C66 , where A — C11C22C33 — C12C23 — C12C33 — C13C22 ^ C12C13C23 .
Physical and technical elastic constants are related by expressions
Ex = ai1 = C11 - (C12C33 + C13C22 - 2C12C13C23 )(C22C33 - C23 ) ' Ey = a21 = C22 (C11C223 + C12C33 - 2C12C13C23 )(C11C33 - C13 ) ' Ez = a33 = C33 (C11C23 + C13C22 - 2C12C13C23 )(C11C22 - C12 ) '
Gyz = a44 = C44; Gxz = a5 51 = G55 ; Gxy = a661 = G661 ; Mxy = a12 ! a22 = (C12C33 - C13C23 )(C11C33 - C13 ) Mxz = a13 ! a33 = (C12C23 - C13C23 )(C11C22 - C12 ) Myx = a12 ! a11 = (C11C33 -C13C23 )(C22C33 - C23 ) №zx = a23 ! a22 = (C11C23 - C13C22 )(C11C33 - C13 ) Myz = a23 ! a33 = (C11C23 -C13C22 )(C11C22 - C12 ) Mzy = a23 ! a22 = (C11C23 C13C22 )(C11C33 - C13 ) ' where Ex , Ey , Ez , are elastic moduli; Gyz , Gxz , Gxy are shear moduli and Vxy , Vxz , Vyx , Vzx , Vyz , ^zy
(3)
are
Poisson's ratios along the directions of elastic symmetry axes.
In the case of a plane stressed state, the elastic constants can be represented in the form [10, 12]
a11 =
a22
С22 (С11С22 С12 ) , С11 (С11С22 - С12 ) ,
(4)
a66 = С66
Then the values of the elastic constants can be represented as:
Ex = C11 - C12 ! C22 ; Ey = C22 - C12 ! C11 ;
Gxy = a66 ; №yx = C12 ! C22 ;
№yx = C12 ! C11
It should be noted that the elastic constants of the composite material can be determined by a nondestructive method by the parameters of the propagation of elastic waves [9, 11].
The above Equations (3) and (4) make it possible to calculate the elastic characteristics of the composite material only along the entire elastic symmetry. To determine the elastic characteristics of orthotropic materials in arbitrary directions, the following expressions are proposed [12, 13]
E^ = Ex (cos4 p + b' sin2 p + Asin4 p) ,
= [^ -0.25(1+ Л -4b')sin2 2p][cos4 p + b'sin2 p + Лsin4 p] 1,
Gp = Gxy [1-(1-d)sin22pj 1,Л = Ex /Ey,b' = 0.25(Л +1),
E
'45
d = (1 + A + 2^ )( 4b'+ 2^ )_1. By analogy with the anisotropic elastic characteristics in [4], expressions describing the anisotropic strength
properties are proposed
C=c(cos4 ( + b sin22p + C sin4 p) .
(5)
where op is the ultimate tensile or compressive strength in an arbitrary direction; o0 is ultimate strength at pure shear at an 450 angle to the axis of elastic symmetry
c = o0 /o90; b = a -0.25 (hc); a = o0 / o45 . (6)
For ultimate strength at pure shear o0 in an arbitrary direction in the x,y plane an expression is suggested.
On the basis of numerous experiments, good convergence of experimental and calculated data obtained by Equations (5) and (6) has been established. Of considerable interest are the conclusions about the strength made on the basis of the generalized Goldenblatt - Kapnov criterion, which has the below form [6-10]
1/ op = nOj cos2 (p + n02 sin2 (p + cos4 (p + n0222 sin4 (p + (n^ + 0.5n0U2) sin2 (p;
rcv = -n^ cos2 (p - n02 sin2 pp + yn0u cos4 (p + n0222 sin4 (p + (n0212
1/ ocv = -n0j cos2 (p - n02 sin2 pp + jtf°n cos4 (p + n0222 sin4 (p + (n^ + 0.5n|l112) sin2 p; 1/ o+ = (n0i + n222 ) sin2 (p + V(n0m - n0222 - 2n0i22) sin2 2( + 4n°02i2 cos 2p; 1/o- = -(-n|1 -n222 )sin2 (p + ;
n?! = 0.5 (1/o(-1/oc); n 02 = 0.5 (1/o£-1/ o9c0); n?,,, = 0.25 (1/op-1/00c);
n2222 = 0.25(1/09p0 -1/09c0);ni0122 = 0.125(1/o0-1/o^)2 +(1/09^0-1/o^)2-(1/r^ + 1/r4-5)2;
ni0212 = 0.0625 (1/T0++1/ )2.
Strength anisotropy according to the Goldenblatt - Kapnov criterion can be used only after strength indicators are experimentally determined.
The dependence of the strength of composite materials on the direction of testing according to [2-5] has the form
o( = o0 ^cos2 (p + a 2 sin2 (p + 0.5 (2b 2 - a 2 -1) sin2 2p ) .
where a = o / o0; b = oAi / o0.
Conclusion
In industry, structures are widely used, where strength determines their performance. Among such structures are hollow bodies of revolution (pipes, tanks, cylinders, etc.). Stress states arise in these structures under the action of operational loads. The peculiarity of the composite material role in the structure is to minimize the stress level perpendicular to the plane of the reinforcing layers. Under uniaxial stress of composite material specimens, the axes of which lie in the plane of reinforcement and make an angle a with the axes of elastic symmetry of the material (in relation to the turns directions) the stresses will be equal (Fig.)
ox = ob cos2 a;oy = ob sin2 a;o = 0.5o6 sin2 2a.
If the degree of a composite anisotropy (o90 / o0 ;o45 / o0) is
known, then it is enough, for a given material, to experimentally determine only one strength characteristic, for example, 60. At that the ultimate strength in any direction is determined by the Equation (5).
Fig. Scheme of the stress state in the For non-destructive strength tests of composite products, the optimal sample under tension at an angle to strength criterion will be that which can be expressed through the the axes of elastic symmetry of the anisotropy index determined directly in the product in different composite material structural directions without destroying them.
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Hamlet Gurgen Shekyan, Doctor of Science (Engineering), Professor (RA, Yerevan) - National Academy of
Sciences of Armenia, Institute of Mechanics, Senior Researcher, hamlet@mechins.sci.am
Norayr Grigor Hovumyan, Doctor of Philosophy (PhD) in Engineering (RA, Yerevan),
newmanhov@yahoo.com
Aramayis Viktor Gevorgyan, Doctor of Philosophy (PhD) in Engineering (RA, Yerevan) - Electromash
GAM Scientific Production and Test Center (CJSC), Director, armenia@giamech.ru
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License
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Received: 16.12.2021 Revised: 10.02.2022 Accepted: 15.02.2022 © The Author(s) 2022
