CALCULATING THE STRENGTH OF A STRINGER PANEL. WITH PROCESS INDUCED DEFECTS AND DEFORMATIONS. As an example of the calculation of ...
ISSN 10526188, Journal of Machinery Manufacture and Reliability, 2014, Vol. 43, No. 1, pp. 36–41. © Allerton Press, Inc., 2014. Original Russian Text © A.E. Ushakov, A.A. Safonov, I.V. Sergeichev, A.Yu. Konstantinov, F.K. Antonov, 2014, published in Problemy Mashinostroeniya i Nadezhnosti Mashin, 2014, No. 1, pp. 46–52.
RELIABILITY, STRENGTH, AND WEAR RESISTANCE OF MACHINES AND STRUCTURES
Modeling ProcessInduced Deformations of Structural Elements Made of Composite Materials A. E. Ushakov, A. A. Safonov, I. V. Sergeichev, A. Yu. Konstantinov, and F. K. Antonov Abstract—A method for estimating the strength of structural elements made of composite materials with a thermosetting matrix, which was characterized by the presence of exfoliationlike defects and processinduced deformations, was proposed. In order to determine the processinduced deforma tions, a mathematical model, which considered thermal and chemical deformations, heat release in the course of matrix polymerization, as well as changes in the matrix properties in transition from the superelastic to the solid state, was implemented. A modeling of the deformation of a standard blank with an interstitial exfoliationlike defect was carried out. A quantitative defect of the effect of the pro cessinduced deformations and the initial dimensions of the defects on the load at which the defect begins to grow was made. DOI: 10.3103/S1052618814010191
INTRODUCTION The lack of a required accuracy for maintaining the parameters of a production process used to man ufacture articles from composite materials, the employment of substandard components, and the effect of a number of random factors lead to the formation of various types of defects in the material of a structure. These defects may cause local deterioration of the physicomechanical characteristics of the material or a wider scatter of their values [1]. The experience of developing and manufacturing composite material parts suggests that exfoliation like defects (unsized regions) and processinduced deformations, which cause the warping of a structure, are the most dangerous defects of loadbearing composite structures. In this work, we carried out an anal ysis of the joint effect of such defects on the strength of structural materials. To this end, we considered a mathematical model of the polymerization of a material with a thermosetting matrix. This model was implemented as a user’s model of the material in the ABAQUS applied software package intended for cal culating residual processinduced deformations and stresses. The static loading of a standard structure with original exfoliationlike defects and residual processinduced deformations was calculated using a numerical implementation of the model of the polymerization of a composite material (CM). PREDICTING SHAPE DISTORTIONS DURING PRODUCTION OF BLANKS FROM CMS WITH THERMOSETTING MATRIX In the polymerization of a material with a thermosetting matrix, e.g., epoxy resin, residual stresses and (or) shape distortions can arise in a blank due to thermal and chemical deformations. One of the basic fac tors that determines the initiation of the residual stresses is the thermal shrinkage of the blank when it is cooled from the polymerization temperature to room temperature. Another important factor is the chem ical shrinkage of the thermosetting matrix in transition from the highly elastic to the solid state. When developing a process for the manufacture of complex shaped CM parts, the behavior of the material with allowance for all abovedescribed peculiarities can only be assessed using numerical methods. In order to obtain a valid result, both heat transfer processes and mechanical processes should be considered. THERMOMECHANICAL MODEL OF BEHAVIOR OF CM WITH THERMOSETTING MATRIX In work [2], the mathematical model for describing the behavior of a CM with a thermosetting matrix in the course of polymerization was proposed; this model considers the following basic processes: thermal and chemical deformations, heat release in the course of matrix polymerization, and changes in the matrix properties in transition from the highly elastic to the solid state. The chemical reaction of the polymerization of the thermosetting material is described by an ordinary differential equation of type dX/dt = f(X, T) where X is the degree of material polymerization, which is 36
MODELING PROCESSINDUCED DEFORMATIONS
37
varied from 0 to 1, and T is the temperature. The extension deformation consists of the thermal εT and the chemical εC components, i.e., t E ε ij
=
T ε ij
+
C ε ij ,
T ε ij
=
∂T
dt'. ∫ α ( T, X ) ∂t' ij
0
The thermal expansion coefficients αij depend on the temperature and the degree of material polymer ization as follows: l
⎧ α ij ⎪ α ij = ⎨ α ijr ⎪ ⎩ α ijg
at
X < X gel
and
T ≥ T g ( X ),
at
X ≥ X gel
and
T ≥ T g ( X ),
at
T < T g ( X ),
where the superscripts l, r, and g correspond to the liquid, highly elastic (amorphous), and glassy (solid) states; Tg is the glass transition temperature; and Xgel is the degree of polymerization at which the material transits from the liquid to the highly elastic state. The chemical shrinkage deformation is introduced as follows: l
t C ε ij
=
∫ 0
β ij ( T, X ) ∂X dt', ∂t'
⎧ β ij , ⎪ β ij = ⎨ β ijr , ⎪ ⎩ β ijg ,
where
X < X gel
and
T ≥ T g ( X ),
X ≥ X gel
and
T ≥ T g ( X ),
T < T g ( X ).
The glass transition temperature at this degree of polymerization is determined from the following equation: T g – T g0 λX = , 1 – ( 1 – λ ) X' T g∞ – T g0 where Tg0 and Tg∞ are the glass transition temperatures for the completely nonpolymerized (X = 0) and the completely polymerized (X = 1) materials, respectively; and λ is the material constant. As the governing relations for the material, we use the following nonlinear viscoelastic anisotropic model [3]: r
E
⎧ C ijkl ⋅ ( ε kl – ε kl ), T ≥ T g ( X ), σ ij = ⎨ g E g r E ⎩ C ijkl ( ε kl – ε kl ) – ( C ijkl – C ijkl ) ⋅ ( ε kl – ε kl )
T < T g ( X ),
t = t υit ,
where tvit is the time of the last transition of the material from the highly elastic to the solid state and Cijkl the relaxation modulus tensor: ⎧ 0, X < X gel , ⎪ P p C ijkl ( t ) = ⎨ ∞ p ⎛ e –t/ρijkl⎞ , + C ⋅ C ijkl ⎝ ⎪ ijkl ⎠ ⎩ p=1
∑
X ≥ X gel .
In order to implement this governing relation in a finiteelement code, the incrementalization proce dure is used, i.e., the following dependence of the increments of the stresses on the increments of the deformations is derived: σ ij ( t + Δt ) = σ ij ( t ) + Δσ ij , ∞ 1 C ijkl = C ijkl + Δξ
E
R
Δσ ij = ΔC ijkl Δ ( ε kl – ε kl ) + Δσ ij ,
P
∑ρ
p p ijkl C ijkl ( 1
– κ ),
R
3
3
P
p
Δσ ij = – Σ k = 1 Σ l = 1 Σ p = 1 ( 1 – κ ) S ijkl ( t ),
p=1
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USHAKOV et al. E
p p p Δ ( ε kl – ε kl ) p S ijkl ( t + Δt ) = κS ijkl ( t ) + ρ ijkl C ijkl ( 1 – κ ), Δξ
3.1 38
p
where Δξ = Δt/aT, κ ≡ exp(–Δξ/ ρ ijkl).
R5
The use of the expansion of the exponent in the Maclaurin series in
x y
80 p
p
p
exp ( – Δξ ω/ρ ijkl ) = 1 – Δt ω/ρ ijkl + O ( Δtω/ρ ijkl ) Fig. 1.
2
and the limiting transition at ω → ∞ yield the following expres sions: ∞
ΔC ijkl
⎧ C ijkl , T ≥ T g ( X ), ⎪ p = ⎨ ∞ p C ijkl , T < T g ( X ), ⎪ C ijkl + ⎩ p=1
∑
l S ij ( t
l
3
⎧ – S I ( t ), T ≥ T g ( X ), R Δσ ij = ⎨ ij ⎩ 0, T < T g ( X ),
3
⎧ 0, T ≥ T g ( X ), ⎪ p + Δt ) = ⎨ l p E C ijkl ( Δ ( ε kl – ε kl ), ⎪ S ij ( t ) + ⎩ p=1
∑
p
T < T g ( X ), )
p
where S ij ( t ) = Σ k = 1 Σ l = 1 Σ p = 1 S ijkl ( t ) . ∞
Associating the complete relaxation modulus tensor C ijkl with the highly elastic state stiffness tensor ∞
r
p
p
g
C ijkl and the nonrelaxed stiffness tensor C ijkl + Σ p = 1 C ijkl with the glassed material stiffness tensor C ijkl , we obtain the following final expressions for calculating the increments of the stresses: ⎧ 0, T ≥ T g ( X ), I S ij ( t + Δt ) = ⎨ I g r E ⎩ S ij ( t ) + ( C ijkl – C ijkl )Δ ( ε kl – ε kl ), r ⎧ C ijkl Δ ( ε kl
Δσ ij = ⎨ ⎩
–
E ε kl )
g C ijkl Δ ( ε kl
–
–
I S ij ( t ),
E ε kl ),
T < T g ( X ),
(1)
T ≥ T g ( X ),
T < T g ( X ).
Relation (1) represents the simplified incremental model, which considers the effect of the loading path, i.e., the history of changes in the state parameters εkl, T, and X. This model was derived using the following simplifications and assumptions: the thermal expansion coefficients in the elastic and solid states are independent of the degree of polymerization X; in the elastic and solid states, the behavior of the r g material is linearly elastic, and the rigidity tensors C ijkl and C ijkl are independent of the degree of poly merization X and the temperature T; in transition from the elastic to the solid state, heating occurs suffi ciently rapidly and, in the reverse transition, it occurs sufficiently slowly to allow us to neglect the velocity effects. The described thermomechanical model was implemented as a user’s subprogram in the ABAQUS applied software package [4]. Since the polymerization of parts made of CMs with a thermosetting matrix is long (tens of hours), the implicit integration procedure was used to obtain a numerical solution. CALCULATING THE STRENGTH OF A STRINGER PANEL WITH PROCESSINDUCED DEFECTS AND DEFORMATIONS As an example of the calculation of the strength of a CM structural element with processinduced defects and deformations, we considered a routine problem of the deformation of a fragment of a plane panel stiffened by Lstringers (Fig. 1). JOURNAL OF MACHINERY MANUFACTURE AND RELIABILITY
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Table Binder State
E, MPa
υ
Solid 2600 0.38 Elastic 2.8 0.497
Composite α, 10–6 1/K
E1, GPa
E2, MPa
G12, MPa
G23, MPa
ν12
ν23
α, 10–6 β1, β = β3 α2 = α 3 1/K ×10–3 2
71 178
168 167
7700 12
4700 42
3100 42
0.28 0.33
0.24 0.24
0.19 –0.25
54 121
–3.6 –0.08
–22 –35
The selection of the stringer profile (the unsymmetrical section) for assessing the effect of the process induced deformations on the evolution of original exfoliationlike defects under external loading is caused by the warping of the stringer in the course of polymerization. The stringer and panel consist of eighteen monolayers with orientation [0, 45, 90, –45, 90, 45, 90, –45, 0]s. DETERMINING CONSTANTS OF THERMOSETTING MATERIAL MODEL The effective characteristics of the monolayers for the two phase states were calculated using the mix ture rule. The equations for the modified mixture rule are written as follows [5]: –1
E 1 = υ f E 1f + υ m E 1m ,
⎛ υf 1– υ⎞ E 2 = ⎜ + f⎟ , Em ⎠ ⎝ υ f E 2f + ( 1 – υ f )E m –1
G 12
⎛ 1– υ⎞ υf = ⎜ + f⎟ , Gm ⎠ ⎝ υ f G 12f + ( 1 – υ f )G m
G 23
⎛ 1– υ⎞ υf = ⎜ + f⎟ , Gm ⎠ ⎝ υ f G 23f + ( 1 – υ f )G m
–1
ν 12 = υ f ν 12f + υ m ν m ,
E2 ν 23 = – 1, 2G 23
where υ is the volume fraction of the phase, and the subscript m indicates the matrix of the composite and the subscript f indicates the reinforcing fiber. Since the characteristics of the matrix LY5052 [6] and those of the CM in the solid state are known, these equations make it possible to calculate the characteristics of reinforcing fibers, such as E1f, E2f, G12f, G23f, and ν12f. Then, using the mixture rule for the fibers and matrix in the liquid state [6], we can calculate the mechanical characteristics of the CM before polymerization. It is known that the thermal expansion coef ficients of carbon fibers are α1f = –0.25 × 10–6 1/K and α2f = α3f = 25 × 10–6 1/K. Using the mixture rule [5], we calculate the effective thermal expansion coefficients for the CM in the two states as follows: α 1 = ( E m ⋅ α m ⋅ υ m + E 1f ⋅ α 1f ⋅ υ f )/ ( E m ⋅ υ m + E 1f ⋅ υ f ), α 2 = α 3 = α 2f ⋅ υ f + ( 1 + ν m ) ⋅ α m ⋅ υ m . The characteristics of the binder and CM for the two phase states are presented in the table. The chemical reaction of polymerization and the resulting internal stresses were calculated using the described model. WARPING CALCULATION The calculation included two stages. At the first stage, a temperature problem was solved. The initial temperature conditions were assigned in accordance with the dependence presented in Fig. 2. The ampli tude and duration of loading were selected based on conditions of real processing of manufacturing arti cles with a thermosetting matrix [6]. JOURNAL OF MACHINERY MANUFACTURE AND RELIABILITY
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USHAKOV et al. T, °C
0.344 0.310 0.275 0.241 0.206 0.172 0.138 0.103 0.069 0.034 0
90 1
70 50
2
30 10 0
5
Y
.
Z
X
3 10
20
30 t, h Fig. 3. Displacements of panel fragment during warp ing, mm (deformation scale is 1 : 20).
Fig. 2. Time dependencies of thermal load in (1) chemical reaction modeling, (2) quasistatic calcula tion of blank in matrix, and (3) static calculation.
M, N m 100
(a)
(b)
80
80 60 60 40 40 20
20 0 70
0 70
(c)
50
50
30
30
10
10
0
0.05
0.10
0.15
0.20 0
(d)
0.05
0.10
0.15
0.20 ϕ, rad
Fig. 4. Dependencies of resultant moments on angles of rotation (solid lines) without and (dashed lines) with warping at various defect dimensions, mm: (a) 5; (b) 10; (c) 15; and (d) 20.
The static calculation stage was then performed to calculate the stressstrain state of the structure under the effect of temperatures and internal stresses accumulated in the blank in the course of polymer ization. The blank deformation after the second stage is shown in Fig. 3. EFFECT OF PROCESSINDUCED DEFECTS ON CRITICAL LOAD In order to assess the effect of the processinduced defects on the stiffness and strength of the panel, we performed modeling of the static loading of the structure. The most critical (for the given problem) case of the loading of the panel fragment, i.e., the bending of the stringer wall outside was considered. For this purpose, one end of the panel was rigidly fixed; at the other end, an angular displacement of 0.2 rad was assigned. As the resultant load, we considered the bending moment, which acted in the section of the upper horizontal side of the stringer wall. JOURNAL OF MACHINERY MANUFACTURE AND RELIABILITY
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In the calculation model, a twodimensional original defect, which simulated the exfoliation region located within the curvature 90 radius that corresponded to the conversion of the stringer into the flange between the 9th and 10th monolayers, was assigned. The 70 dimensions of the defect along the stringer and the curvature were 1 specified the same, i.e., 5–20 mm. 50 In order to calculate the defect growth initiation load, the vir 30 2 tual crack closure criterion [7] was used in which the equivalent fracture toughness was expressed as a power law [8] with the 10 parameters GIC = GIIC = GIIIC = 168 N/m and am = an = a0 = 1. 1.5 7.5 12.5 17.5 0 Figure 4 shows the dependencies of the resultant moment on Size of defect, mm the angle of rotation. Fig. 5. Dependencies of defect The effect of the defect dimensions and the processinduced growth initiation load on initial deformations on the strength of the stringer panel fragment under defect size (1) without and (2) with consideration can be assessed from the dependencies of the criti warping. cal value of the moment (the defect growth initiation load) with and without the processinduced deformations (Fig. 5). It can be seen in the curves presented in Figs. 4 and 5 that the residual processinduced deformations can substantially reduce the defect growth initiation load. With an increase in the initial defect dimen sions, the negative effect of the processinduced deformations becomes much more pronounced; for example, a 20mmdiameter exfoliation defect leads to a loss of the loadcarrying capacity of the structure of almost 50% under the given loading mode. It is recommended that the developed model be used to predict optimum process conditions, e.g., the tool temperature, the exposure time at various process stages, the tool lead, etc. in order to minimize residual processinduced stresses, which result in the origination and growth of processinduced defects. MС, N m
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Translated by D. Tkachuk
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