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Rizov V.I. Utilization of viscoelatic models with non-linear springs and dashpots in delamination study of multilayered beams. PNRPU Mechanics Bulletin, 2022, no. 1, pp. 5-13. DOI: 10.15593/perm.mech/2022.1.01
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ВЕСТНИК ПНИПУ. МЕХАНИКА № 1,2022 PNRPU MECHANICS BULLETIN
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Research article
DOI: 10.15593/perm.mech/2022.1.01 UDK 531
UTILIZATION OF VISCOELATIC MODELS WITH NON-LINEAR SPRINGS AND DASHPOTS IN DELAMINATION STUDY OF MULTILAYERED BEAMS
V.I. Rizov
University of Architecture, Civil Engineering and Geodesy, Sofia, Bulgaria
ARTICLE INFO
ABSTRACT
Received: 11 November 2021 Approved: 29 March 2022 Accepted for publication: 01 April 2022
Keywords:
inhomogeneous material, multilayered beam, non-linear springs and dashpots, delamination, non-linear viscoelastic behavio.
This paper analyzes delamination of multilayered inhomogeneous beam structure by utilization a non-linear viscoelastic mechanical model. The non-linear time-dependent response is treated by a non-linear spring and a non-linear dashpot connected in series to a system of two linear springs and two linear dashpots. The model is under stress that is a linear function of time. The constitutive law of the model representing a non-linear dependence between stress, strain and time is derived. The main goal of the paper is to obtain a solution of the strain energy release rate for the delamination in the multilayered inhomogeneous beam by applying the nonlinear viscoelastic model. Solutions are derived by using the complementary strain energy and by analyzing the balance of the energy with taking into account the non-linear viscoelastic behaviour. For this purpose, the constitutive law of the non-linear viscoelastic mechanical model is used. The solutions are applied to obtain results for multilayered beams with non-linear variation of material properties in longitudinal direction. The influence of different parameters on the time-dependent strain energy release rate is assessed. The study indicates the effectiveness of the viscoelastic mechanical models with non-linear springs and dashpots in time-dependent delami-nation analyses of multilayered inhomogeneous beam structures.
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Introduction
In contemporaneous engineering practice very often load-bearing engineering structures are subjected to heavy loadings and extreme external influences. At the same time, all the requirements towards the engineering structures have to be satisfied with minimum expenditure of materials not only because of the cost but also because of the fact that the weight of structures is usually an issue. This situation leads to necessity of introduction of new structural materials of adequate properties to withstand the
complicated conditions of work. One type of such materials is continuously inhomogeneous structural materials [1-12]. One of the most important peculiarities of these materials is the fact that their properties vary smoothly along one or more coordinates in the structural member [13-15]. In the last four decades, the most widely used continuously inhomogeneous materials are the functionally graded materials [16-23]. They represent continuously inhomogeneous composites made by mixing of two or more constituent materials. The microstructure and the properties of the functionally graded materials can be
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formed technologically during the production process [24-34]. In this way, highly sophisticated structural members with different material properties in different parts of a member can be manufactured.
Multilayered inhomogeneous materials representing systems of adhesively bonded layers are frequently applied in various load-bearing structures in modern engineering [35-44]. Structural members made of multilayered materials have various advantages in comparison with the homogeneous load-bearing structures. For instance, the strength-to-weight and stiffness-to-weight ratios of multi-layered systems are higher. Although the concept of multi-layered materials is rather efficient, the multilayered structural members are highly sensitive to delamination (separation) of layers [45-48]. Delamination can grow rapidly causing the failure of the multilayered inhomogeneous structure with catastrophic consequences. There are various factors which affect the delamination behaviour. The effects of these factors have to be analyzed at length in order to improve the knowledge on the delamination. This knowledge can play an important role in safety design of load-bearing multilayered structural components.
Usually, the multilayered inhomogeneous materials have non-linear viscoelastic behaviour. Therefore, non-linear vis-coelastic mechanical models have to be utilized in time-dependent delamination analyses of multilayered structures.
For this purpose, in the present paper, a model representing a combination of linear springs and dashpots is modified by introducing a non-linear spring and a nonlinear dashpot. In this manner, a non-linear viscoelastic model is obtained. This model is utilized in a delamination analysis of a multilayered inhomogeneous beam. A solution of the strain energy release rate is derived. The balance of the energy is analyzed to verify the solution. When deriving the solution, the constitutive law of the non-linear viscoelastic model is used to describe the mechanical behaviour of the beam structure. It should be noted that this paper develops further the approach presented in [49; 50]. The main novelty is that a non-linear viscoelastic model is utilized in contrast to [49; 50] where delamination is analyzed by using viscoelastic models assembled by linear springs and linear dashpots.
Analysis of the time-dependent strain energy release rate
The mechanical model presented in Fig. 1 exhibits nonlinear viscoelastic behaviour due to the non-linear spring, npt, and the non-linear dashpot, nwi, which are added to the linear springs with modulae of elasticity, EBi and EDi, and linear dashpots with coefficients of viscosity, nBi and nDi.
The model in Fig. 1 is under stress, ai, that varies with time, t, according to following equation:
a. = vt, (1)
i ai ' V /
where vai is a parameter that governs the loading.
Fig. 1. Non-linear viscoelastic model
The constitutive law of the model represents a nonlinear dependence between stress, strain and time (the non-linearity is generated by the non-linear spring and dashpot). In order to derive this constitutive law, first, the strain, ei, is expressed as (Fig. 1)
8 i = 8 Qi + 8 r
(2)
where eQi is the strain in the spring with modulus of elasticity, EBi, and in the dashpot with coefficient of viscosity, nBi. The strain, e Rl, is written as
8R = 8 . + 8
Ri npi nwi
(3)
where enpi is the strain in the non-linear spring, enwi is the
strain in the non-linear dashpot as shown in Fig. 1.
The mechanical behaviour of the non-linear spring is described by the following non-linear stress-strain relationship [51]:
Hi8 npi
(4)
+ 8 i
where anpi is the stress in the spring, is a material property. From Fig. 1, it is obvious that
a . = a.. (5)
npi i
After substituting of (5) in (4) and performing some mathematical transformations, one derives
(6)
VH^O"'
The response of the non-linear dashpot is treated by using the following relationship:
L 8 n
Vi+f 28nwi'
(7)
where Li and f are material properties, enwi is the first derivative of the strain with respect to time. By using (7) and taking into account that (Fig.1)
a
a
8
g =
one derives
By substituting of (1) in (9), one obtains
vj
JLFÏ2
(8)
(9)
(10)
Equation (10) is solved with respect to enwi. The result is
(ii)
Ji Vai
where Ci is an integration constant. Since
e .(0) = 0.
nwi \ } -
one derives
A f
C =—i—
1 A. ■
(12)
(13)
By combining of (11) and (13), the strain in the nonlinear dashpot is found as
e =
nwi r2_
J i
V- [ L-JLJf |. (14)
The next step is to obtain the strains in the linear springs and dashpots. For this purpose, the following equation of equilibrium is used:
(15)
where o„ , c E and c„ are the stresses in the spring with
'iDi EDi %i a ^
coefficient of viscosity, nD1, in the spring with modulus of elasticity, EDj, and in the dashpot with coefficient of viscosity, nBj, respectively (Fig. 1). These stresses are expressed as
= eQi nDi,
°EB, EDieQi ■
= ПйеBi :
(16)
(17)
(18)
where eBi is the first derivative of the strain in the dashpot with coefficient of viscosity, nB1 • The stress in the spring with modulus of elasticity, EBj, is equal to c
EBi (eQi eBi ) nBieBi •
By using of (15)-(19), one obtains
eBi + VueBi + V2ieBi = ¥2/ :
where
(19)
(20)
Vu = ^, V2i=Et , V2i=Er, V3i =
6ц. Ьц Ьц Ьц
s,i = ^^, b2i = nri + + nBi. (21)
Ed
F F
Bi Bi
The solution of (20) is found as
eB¡ (t) = A - Bip2iep + Bip1i - A e^ +A,.t + b. , (22)
P2i - P11
P2i - Pb
where
V3i
Vuva,
A = —, B . =-Vfr, Pu =-0.5Vu +Pxi,
V2 i V2 i 1i
Pxi = 0.5(v2 - 4V2i)0 5 , P2i =- 0.5VU -Pxi. (23) By combining of (1), (19) and (21), one derives
°i a Vu ,
e Qi (t ) = a. V"1' + a ^2ieP2t + -
s1i "i V2is1it
nBiai I P 1 eP„t + P eP2,t +__
but
Pu+P2i^2/^+
(24)
where
x = v2i+v„- x = -Vu Pi- vl 1i (P2i - P1i )V 2i (P2i - Pu )v2iV'
x = ¥2i +¥ü 1i ( p2i -p1i )51it'
^2i =
-V1iPi - V2i (P2i - P1i ) V
(25)
(26)
By using of (1), (2), (3), (6), (14) and (24), one obtains the following non-linear constitutive law that relates time, stresses and strains in the model in Fig. 1:
£i (t) = Gi V* + °i ^eP2t + - g, +
51i -2i51it
Пв. ai
Eb.
t, 1
Pu Vм + P2iV*' + — I +
s1it J jnf-a,
ж ( l J. <27)
The constitutive law (27) is applied for treating the time-dependent mechanical behaviour of the i-th layer in the multilayered inhomogeneous non-linear viscoelastic beam structure in Fig. 2 when deriving a solution of the strain energy release rate.
The beam consists of an arbitrary number of adhesively bonded layers with different thicknesses and material properties. The cross-section of the beam is a rectangle of width, b , and thickness, h . The beam length is denoted by l. The beam is clamped in its right-hand end. A delamination crack
a
Fig. 2. Multilayered beam structure with delamination crack
of length, a, is located between layers so as the thicknesses of the lower and upper delamination crack arms are h* and h2, respectively. The beam is loaded by a bending moment, M, applied at the free end of the upper crack arm. Therefore, the lower crack arm is free of stresses. The bending moment varies with time according to the following dependence:
M _ vt,
(28)
where v is a parameter that controls the loading.
The layers exhibit continuous material inhomogeneity along the beam length. The following dependences are used for describing the variation of the material properties involved in the constitutive law (27) in the longitudinal direction in the z-th layer of the beam:
E _E
E _ E . Brni Bai „Ц1, ^Bi Bai ■■ Л
Г1.
Лл _ nBai +
E _e
E _ E 1 Dmi Dai „Иг. ^Di Dai ■■ Л
l
nBmi _ n l
Bai Цэ,
(29)
nDrni nDai „Ц4, j _ H j ^ ^ J"L ai x^
H„_. _ H„
l Ц4.
i ai
l Ц5,
where
L. _ L . + Lmi Lai хЦ6'
i ai l Ц6,
0 < x < l, i _ 1, 2, ..., m.
(30)
(31)
(32)
In formulae (29) and (30), EBai , EDai , nBai , ^Dai , Hai
and Lai are the values of EBj, EDj, nBi, nDi, H and Lt at the free of the beam, respectively. The values of EBj, EDj, nBi, nDi, Hi and Lt at the clamped end of the beam are
nBmi, nL
Hmi and Lm¡, respectively. The pa-
rameters, , , , ц4,., д5,. and control the variation of EBj, EDj, nBj, nDj, Hi and Ц, respectively, x is the longitudinal centroidal axis of the beam, m is the number of layers.
In order to derive solution of the strain energy release rate for the delamination in the beam in Fig. 2, first, the complementary strain energy in the beam, U *, is obtained as
U _ Uj + U 2
(33)
where U* and U* are the complementary strain energies in the upper delamination crack arm and in the un-cracked beam portion, 0 < x < l, respectively. The complementary strain energy, U* , is found as
i_m1 a zii+i
Uj*_ ^ bJ J u*0idxdzl,
i_1 0 z„
(34)
where m1 is the number of layers in the upper crack arm, u *0¡ is the complementary strain energy density in the i-th layer, z. and z1i+1 are, respectively, the coordinates of the upper and lower surfaces of the layer, z. is the vertical centroidal axis of upper crack arm cross-section.
Due to non-linear character of the constitutive law (27), the stress can not be determined explicitly. Therefore, the complementary strain energy density that is involved in (34) is obtained as
a i
u*. = J e.da.. (35)
0
By combining of (27) and (35), one derives
u' XePlt + X2„eP2t + — _ Vu
0i 2 1 1
Sii vLSi/
nBiPi " 2EB¡
Pii V*1' + P2i + J | +
+h, _VHr^ir + L
i
f ч
-siL2 _ fV +arcsin^
^ * i 1 -f T
2 f
(36)
The boundaries of integration, zii and z1i+1, in (34) are
replaced with ami and a^, respectively (here, ami and a^
are the stresses in the upper and lower surfaces of the layer, respectively)
i_m1 a
U*_ ^ bJ J u*0idxdz1.
(37)
i_1 0
Besides, dz1 that is involved in (37) is expressed as a function of ai. For this purpose, the distribution of strains along the thickness of the upper crack arm is written as
e = K1 (z1 - z1n ) (38)
where k1 is the curvature of the crack arm, z1n is the coordinate of the neutral axis. Formula (38) follows from the Bernoulli's hypothesis for plane sections (this hypothesis is applicable here since beams of high length to thickness ratio are under consideration).
From (27) and (38), one obtains
"u ' V2Â/
0 + O ^eP2' + _ +
O,'
nBiCi
E„1
' , 1
Pu\ie + P2i + ^ I +
V J jHf_0
+1 L
By using (39), one derives
2o2
+ zlB . (39)
dz1 = —
ep1t +i eP2< + J__^__
KeP1' + X2i'eP2' + :
+E-1 Pu V1' + P2i V*' + -r | +
№ _ c2 )3 ^
-1i' J JHTO,
do. (40)
_ h V
The quantities, cwi, , k and z1n, are found in the
following way. First, the equations of the equilibrium of the elementary forces in the cross-section of the upper crack arm are written as
i=m1 i=m1
N = £ b J aidz1, M = £ b J aiz1dz1, (41)
i=1 W i=1 W
where N is the axial force in the upper crack arm (for the beam under consideration, N = 0), z1 and dzl are found by (39) and (40), respectively. By using (27) and (41) and by replacing of z1 with zh. and z1i+1, one obtains
K (zu _Z1„) = a^/1' + aroiX2,eP2' _o,
P2' + _ p _
5 roi 2 c ^ 1 V2Â/
+
i
EKi
PuVP1i' + P2,X2,eP2ii + -L | +
§1i' j thfp
^ 1 L )■
fi
K (z1i+1 _ zln ) = awVP1' + a»;VP2' + -2L _a
(42)
5u ¥2i§1i'
E
Jêi
p^' + P2iVP2i' + -L ) +
§1i' J si H
2i
fi
L _
where
i = 1,2,..., m1.
(43)
(44)
It should be mentioned that (42) and (43) are written by replacing of ai with aroi and aéi in (41), respectively.
Equations (41), (42) and (43) are solved with respect to K1, z1n, aroi and odi where i = 1,2,..., m1 by using the MatLab computer program at various values of time.
Formula (37) is applied also to determine U* . For this purpose, m , arot, odi, m*, and z1 are replaced with m,
, , «0® and z2, respectively. Here, and
are the stresses in the upper and lower surfaces of the i-th layer in the un-cracked beam portion, respectively, w*^ is the complementary strain energy density, z2 is the vertical centroidal axis of the cross-section of the un-cracked beam portion. Besides, the boundaries of integration 0 and a are replaced with a and l, respectively. Analogical replacements are performed in equations (41), (42) and (43) and then these equations are used to determine k2 , z2n, aro ^
and a^, (here, k2 is the curvature of the un-cracked beam
portion) at various values of time.
The strain energy release rate, G, for the delamination in Fig. 1 is found as
G =
dU bda
(45)
By substituting of U* and U* in (33) and then in (45),
one derives
m1 atfi
i =m
G =X J M*idz1 J ukidz2
(46)
i
where the stresses and the complementary strain energy densities are found at x = a. The integration in (46) is car-ried-out by using the MatLab computer program. The strain energy release rate is determined by (56) at various values of time.
The strain energy release rate is derived also by analyzing the balance of the energy for verification. The balance is written as
M =dU §a + Gb5a, da
(47)
where 9 is the angle of rotation of the free end of the upper crack arm, 5a is a small increase of the delamination length, U is the strain energy in the beam. From (47), one obtains
G = 1 i MdU
b 1 da da The angle of rotation is found as
a l
9 = J K1dx +J K2dx .
(48)
(49)
0 a
2
a
mi
It should be noted that (49) is derived by using the integrals of Maxwell-Mohr. The strain energy in the beam is written as
U = U, + U 2,
(50)
where U1 and U2 are the strain energies cumulated in the upper crack arm and in the un-cracked beam portion, respectively. The quantity, U1, is found as
i=m, a a»i
U,= ^ b J! u0idxdz,,
l=l 0 aroi
(51)
where u0i is the strain energy density in the i-th layer. The quantity, u0i, is determined as
0i =aiei _| £idai .
(52)
By using of (27) and (52), one derives
1
u0i = 2
Пв.аг
a;
a-V* +a2^_a2 ¥li
Sli 1 ¥ 2. Si.'
2E
Pl. V1' +P21X 2.ep2lt + —
а
f2
S,t J jHj _a2
-H + JH2 _a2 _L +
' 1 f
f a.
—■Jl2 _ f 2a2 +-'L-arcsin л 2 f
jA
L.
(53)
The strain energy in the un-cracked beam portion is found by replacing of m1, ora i, odi, u0i and z1 with m ,
, , u0^. and z2 in (51). By combining of (48), (49), (50) and (51), one obtains
G = 1
b
1=m, "vi
M (K1 _K2 )- bI J u0idz,
+ J u0^idz2
(54)
where the curvatures, the stresses and the strain energy densities are found at x = a. The integration in (54) is carried-out by the MatLab computer program. The strain energy release rate is determined by (54) at various values of time. It should be noted that the strain energy release rates found by (54) are exact matches of these obtained by (46) which is a verification of the solutions.
Numerical results
This section deals with numerical results obtained by applying the solution of the strain energy release rate derived in
section 2 of the paper. The numerical results illustrate the effects of various parameters on the delamination of the mul-tilayered inhomogeneous non-linear viscoelastic beam. The strain energy release rate is presented in non-dimensional form by applying the formula GN = G / (EBalb) .
Fig. 3. Two three-layered beam structures (a) with delamination
between layers 1 and 2, and (b) with delamination between layers 2 and 3
Two three-layered beam configurations are considered in order to investigate the influence of the delamination crack location along the beam thickness (Fig. 3). In the beam depicted in Fig. 3, a, the delamination is located between layers 1 and 2. A beam with delamination between layers 2 and 3 is also considered (Fig. 3, b). The layers in the beams in Fig. 3 have identical thicknesses. The beams are clamped in their right-hand ends. It is assumed that b = 0.025 m, h = 0.012 m, l = 0.400 m, f = 1 s and v = 0.9 • 10-7 Nm/s.
First, the variation of the strain energy release rate with time is analyzed. The beam with delamination between layers 1 and 2 is considered. The results obtained are shown in Fig. 4 where the strain energy release rate is plotted against the non-dimensional time (the latter is obtained by using the formula tN = tEBa1 / nBa1 ). The curve in Fig. 4 indicates that the strain energy release rate increases with time.
This is due to two factors (increase of the external loading with time and the viscoelastic behaviour of the multi-layered beam).
The variation of the non-dimensional strain energy release rate with EBra1 / EBa1 ratio for both three-layered beam structures is shown in Fig. 5.
It is apparent from Fig. 5 that the strain energy release rate decreases with increasing of EBra1 / EBa1 ratio.
It can be concluded also that the strain energy release rate for the beam with delamination between layers 1 and 2 is higher than that for the beam with delamination located between layers 2 and 3 (Fig. 5).
t
t
1=1
а
ro
i=l
а
roi;
Fig. 4. Variation of the non-dimensional strain energy release rate with non-dimensional time
Fig. 6. Variation of the non-dimensional strain energy release rate with EDm1 / EDa1 ratio (curve 1 - at a /1 = 0.25 , curve 2 -
at a /1 = 0.50 and curve 3 - at a /1 = 0.75 )
Fig. 5. Variation of the non-dimensional strain energy release rate with EBra1 / EBa1 ratio (curve 1 - for the beam with delamination between layers 1 and 2, curve 2 - for the beam with delamination between layers 2 and 3)
Fig. 7. Variation of the non-dimensional strain energy release rate with nBmJ n Ba1 rati° (curve 1 - at nDm1 / nDa1 = 05 , curve 2 -
at nD»1 / nDa1 = 15 and 3 - at nD»1 / nDa1 = 2-5 )
Figure 6 shows the variation of the non-dimensional strain energy release rate with EDa1 / EDa1 ratio for three a /1 ratios.
The beam with delamination between layers 1 and 2 is analyzed.
From Fig. 6, it can be seen that the increase of the EDm1 / EDa1 ratio leads to decrease of the strain energy release rate. It can also be seen that the strain energy release rate deceases with increasing of a /1 ratio (Fig. 6).
The effects of nBaJ nBai and ^d»i/ ^Dai ratio on the strain energy release rate is illustrated in Fig. 7. It is observed that the strain energy release rate decreases with increasing of nBm1 / ПВа1 ratio (Fig. 7). The effect of
nDra1 / nDa1 ratio is identical.
The variation of the non-dimensional strain energy release rate with Hw1 / Ha1 ratio for three values of the parameter v is displayed in Fig. 8.
The effect of Hra1 / Ha1 ratio is to decrease of the strain energy release rate. From Fig. 8, it can be found also that the strain energy release rate increases with increasing of v.
The influences of Lw1 / La1 and La2 / La1 ratios on the strain energy release rate are also studied. The strain energy release rate increases with the increase of these ratios as shown in Fig. 9.
Conclusion
A delamination analysis of a multilayered inhomogene-ous beam by using a non-linear viscoelastic model is developed. A non-linear spring and a non-linear dashpot are added in a model that consists of two linear spring and two linear dashpots in order to describe the non-linear mechanical behaviour of the beam under consideration. The responses of the non-linear spring and dashpot are treated by applying non-linear constitutive relationships. The stressstrain-time constitutive law of the model is derived. The law is non-linear as a result of adding of non-linear spring and dashpot in the model. One of the important peculiarities of this constitutive law is that the stress can not be determined explicitly, in contrast to constitutive laws of linear viscoe-lastic models. The material properties which are involved in
Fig. 8. Variation of the non-dimensional strain energy release rate with Hm1 / Ha1 ratio (curve 1 - at v = 0.3 • 10-7 Nm/s, curve 2 -
at v = 0.6 • 10-7 Nm/s and curve 3 - at v = 0.9 • 10-7 Nm/s)
Fig. 9. Variation of the non-dimensional strain energy release rate with L^ / La1 ratio (curve 1 - at La2 / La1 = 0.5 , curve 2 -
at La2 / La1 = 1.0 and curve 3 - at La2 / La1 = 2.° )
the constitutive law are continuous functions of the longitudinal coordinate since the layers exhibit continuous material inhomogeneity along the beam length. Solutions of the strain energy release rate for the delamination are obtained by using the complementary strain energy in the beam and by considering the balance of the energy. The two solutions generate identical results which prove their correctness. The solutions are time-dependent and take into account the nonlinear viscoelastic behaviour of the beam under external bending moment that is a linear function of time. The solutions are applied to obtain numerical results which clarify the influence of various factors on the strain energy release rate. The results obtained indicate that the strain energy release rate increases with time. The influence of EBra1 / EBa1,
EDrn1 / EDa1 , ПВю1 / ПВа1 , ^Drnl / ^Dal , Hrn1 / Ha1 , Lrn1 / La1
and La 2/ La1 ratios on the strain energy release rate is
analyzed (these ratios describe the material inhomogeneity along the beam length). The analysis indicates that the strain energy release rate decreases with increasing of these ratios. The effect of a /1 ratio is analyzed too (this ratio represents the relative length of the delamiantion crack). It is found that the strain energy release rate decreases with increasing of a /1 ratio. The results obtained demonstrate that the viscoelastic models with non-linear springs and dashpots are effective tools for treating time-dependent mechanical behaviour of multilayered inhomogeneous beam structures with delaminations when analyzing the strain energy release rate. It should be mentioned that these models can be developed further by introducing of more sophisticated combinations of non-linear springs and dashpots for representing of more complicated time-dependent behaviour of multilay-ered inhomogeneous beams with delamination.
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Финансирование. Исследование не имело спонсорской поддержки. Конфликт интересов. Автор заявляет об отсутствии конфликта интересов.
Financing. The study was not sponsored.
Conflict of interest. The authors declare no conflict of interest.
