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Effect of elastic properties dependence of the stress state in composite materials

  • A. N. Fedorenko
  • B. N. FedulovEmail author
Original Paper
  • 49 Downloads

Abstract

Buckling problem is quite important for the engineering practice, mostly in cases of lightweight structures. The use of composites reduces the weight of the structure but gives problems in choice of reliable methods to model buckling and postbuckling behavior of details, especially if they contain thin-walled components. One of the problems is the choice of correct elastic properties of composite material for analysis. Elastic characteristics of composite material depend on the type of loading and uniaxial tension/compression test results in some cases demonstrate an essential difference. The buckling analysis usually assumes compression of the structure and the choice of elastic constants obtained in compression tests leads to more accurate results, but does not guarantee a good correlation with experiments in case of postbuckling analysis due to ignoring some regions with tension stress state. A possible way of material stress-state sensitivity consideration is usage of material models that take into account stiffness susceptibility to different types of loading. Further development of nonlinear elastic model in problems related to buckling and postbuckling analysis for composite materials up to failure is the introduction of material progressive degradation model to consider material properties reduction due to damage in conjunction with nonlinear elasticity.

Keywords

Composite laminate Nonlinear elasticity Thin-walled structure Buckling 

1 Introduction

One of the most common example of composite material properties with dependency from stress state is different elastic moduli, determined from compression and tension tests. Another important example is nonlinear diagram for shear loading [1]. Understanding of these phenomena is important in engineering practice, for instance, to achieve correlation with experimental loading diagram and deflections of structure under complex loading. Consideration of variable elastic properties is essential for buckling and posbuckling analysis of thin-walled composite structures. Since classic elastic model is unable to capture these effects, a number of nonlinear material models were developed [2, 3, 4]. Considering model with variable elastic properties, it is important to satisfy fundamental principles of continuum mechanics, such as existence of elastic potential and positive semi-definite of corresponding quadratic form. The presented constitutive relations satisfy the above-mentioned requirements and describe simultaneously two types of physical nonlinearities: the nonlinearity of shear stress–strain dependency and the stress state susceptibility of material properties. Developed model is implemented to finite-element solver, and for few test problems it demonstrated much better correlation with experiment than linear elastic model.

2 Anisotropic elastic model susceptible to stress state and nonlinear shear

Nonlinear elastic model is based on the formulations described in [2]. Stress triaxiality parameter \( \xi \) is used to formalize stress state:
$$ \xi = \sigma /\sigma_{0} , $$
where \( \sigma = 1/3\sigma_{ii} \) is the hydrostatic stress component, \( \sigma_{0} = \sqrt {3/2S_{ij} S_{ij} } \) is the von Mises equivalent stress, and \( S_{ij} = \sigma_{ij} - \sigma \delta_{ij} \) is the stress deviator components.
For the development of nonlinear shear material model, the parameter that represents the degree of shear stresses or deformations is formulated in the following form:
$$ Q = D_{ij} \sigma_{ij} , $$
(1)
where \( D_{ij} \) tensor has the following representation in coordinate system coincident with the orientation of anisotropy axes:
$$ D_{ij} = \left[ {\begin{array}{c@{\quad}c@{\quad}c} 0 & {1/2} & 0 \\ {1/2} & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } \right].$$
(2)
Elastic potential can be represented in the form dependent on stress triaxiality and shear parameter as shown below:
$$ \varPhi = \frac{1}{2}A_{ijkl} \left( {\xi , Q} \right)\sigma_{ij} \sigma_{kl} . $$
(3)
In case of plane stress conditions, the value of the parameter \( \xi \) is limited \( - 3/2 \le \xi \le 3/2 \) and the constitutive equations obtained on the base of potential Eq. 3 can be represented in the following form:
$$ \begin{aligned} & \varepsilon_{11} = A_{1111} \left( \xi \right)\sigma_{11} + A_{1122} \left( \xi \right)\sigma_{22}\\ &\qquad\quad + \left[ {\left( {\frac{1}{3\xi } + \frac{3}{2}\xi } \right)\sigma - \frac{3}{2}\xi \sigma_{11} } \right]\varPhi_{1} \sigma_{0}^{ - 2} , \hfill \\ & \varepsilon_{22} = A_{1122} (\xi )\sigma_{11} + A_{2222} \left( \xi \right)\sigma_{22}\\&\qquad\quad + \left[ {\left( {\frac{1}{3\xi } + \frac{3}{2}\xi } \right)\sigma - \frac{3}{2}\xi \sigma_{22} } \right]\varPhi_{1} \sigma_{0}^{ - 2} , \hfill\\ & \varepsilon_{12} = \left[ {\left( {A_{1212} \left( {\xi ,Q} \right) + \frac{1}{2}\frac{{\partial A_{1212} \left( {\xi ,Q} \right)}}{\partial Q}} \right) - \frac{3}{2}\xi \varPhi_{1} \sigma_{0}^{ - 2} } \right]\sigma_{12} , \hfill \\ & \varPhi_{1} = \frac{1}{2}\left[ A^{\prime}_{1111} \left( \xi \right)\sigma_{11}^{2} + A^{\prime}_{2222} \left( \xi \right)\sigma_{22}^{2}\right. \\ &\left.\qquad\quad + 2A^{\prime}_{1122} \left( \xi \right)\sigma_{11} \sigma_{22} + A^{\prime}_{1212} \left( \xi \right)\sigma_{12}^{2} \right], \hfill \\ \end{aligned} $$
(4)
where prime denotes the derivative with respect to parameter \( \xi \) and, according to Eqs. 1 and 2, the dependency on parameter \( Q \) remains only in \( A_{1212} \left( {\xi , Q} \right) \).
Using polynomial substitution \( A_{1212} \left( {\xi , Q} \right) = \mathop \sum \nolimits_{n} C_{n} \left( \xi \right)Q^{n} \) to expression for \( \varepsilon_{12} \) in Eq. 4 and assuming \( B_{n} \left( \xi \right) = \left( {1 + \frac{\text{n}}{2}} \right)C_{n} \left( \xi \right), \) one can obtain
$$ \varepsilon_{12} = \left[ {B\left( {\xi ,Q} \right) - \frac{3}{2}\xi \varPhi_{1} \sigma_{0}^{ - 2} } \right]\sigma_{12} , $$
(5)
where \( B\left( {\xi ,Q} \right) \) is an arbitrary function that can be approximated by polynomial dependency.

The determination of functional dependencies of coefficients \( A_{ijkl} \left( \xi \right) \) is a quite complex problem. Consideration of Eq. 4 with known stress and strain components from one loading test with a certain constant \( \xi = \xi_{0} \) gives the relations between \( A_{ijkl} \) at \( \xi_{0} \). Collecting experimental data for different values of \( \xi_{0} \), one can suppose a form of functional dependencies \( A_{ijkl} \left( \xi \right) \) for the best matching of loading diagrams. A detailed methodology is described in [2]. For the beginning, the cases of uniaxial tension, uniaxial compression and shear loading are considered due to a significant simplification of Eq. 4 for \( \xi = \pm \;1/3 \) and \( \xi = 0. \) Then, a good approach is consideration of proportional biaxial loading in the directions of principal axes of anisotropy. Different loading in off-axis direction of composite material can also be used for verification.

In practical way, the piecewise linear functions for \( A_{ijkl} \left( \xi \right) \) could be used, but following the idea to keep less number of parameters in the model one can define
$$ \begin{aligned} A_{1111} \left( \xi \right) = a_{11}^{0} + c_{11} \xi , \hfill \\ A_{2222} \left( \xi \right) = a_{22}^{0} + c_{22} \xi , \hfill \\ A_{1122} \left( \xi \right) = a_{12}^{0} + c_{12} \xi , \hfill \\ \end{aligned} $$
(6)
with constant \( a_{ij}^{0} \) and \( c_{ij} . \)

The coefficients \( A_{ijkl} \) in general case are not arbitrary. In particular, coefficients have to guarantee positive definiteness of the potential Eq. 3. Analytical solution for finding restrictions for coefficients in linear relations Eq. 6 is still not obtained, but positive definiteness of potential for predefined coefficients can be verified numerically for the range of \( \xi \) between − 2/3 and + 2/3 in the case of plane stress conditions.

In case of pure shear, \( \xi = 0 \) and Eq. 5 can be reduced to expression \( \varepsilon_{12} = B(0,Q)\sigma_{12} \). Since \( B(0,Q) \) assumes the polynomial form, it can be taken in the form proposed by Tsai and Hahn [4]:
$$ \varepsilon_{12} = \frac{1}{2}\left[ {\frac{1}{G} + \alpha \sigma_{12}^{2} - 3\xi \varPhi_{1} \sigma_{0}^{ - 2} } \right]\sigma_{12} , $$
(7)
where G is a constant shear modulus on initial stage of deformation that is commonly used for linear elastic relations and \( \alpha \) is a coefficient which can be defined from shear loading diagram.

It should be noted that Eq. 7 assumes simplification of independence on stress triaxiality: \( B\left( {\xi ,Q} \right) = B(Q) \).

3 Composite cylindrical shell problem

3.1 Description of experiment

Experimental results of Bisagni’s research [5] were used as an example of detailed test data with extensive measurements. This work contains experimental data obtained from buckling and postbuckling tests performed until the collapse of stiffened composite cylindrical shell. According to the experimental data, the tested cylinder has internal diameter and an overall length of 700 mm, including two tabs provided at the top and at the bottom surfaces for fixing the shell into the test equipment. The cylinder is rigidly fixed through end tabs and loaded to compression applying displacement up to failure. The actual length is, therefore, limited within the central part of the cylinder height and is equal to 540 mm. The shell is reinforced by eight L-shaped stringers, parallel oriented and equally spaced in circumferential direction. The blade of the stiffeners is 25 mm long, while the flange attached to the skin of the cylinder is 32 mm long. The cylinder dimensions are presented in Table 1. The shell of carbon fiber reinforced plastics (CFRP) is made of fabric tape with material properties shown in Table 2, as given in [5].
Table 1

Stringer-reinforced cylinder characteristics

Parameter

Value

Shell diameter (mm)

700

Shell length (mm)

700

Stringers

 Number

8

 L length (mm)

700

 L width (mm)

25×32

Lay-up

 Skin

[+ 45°/− 45°]

 Skin (reinforcements)

[+ 45°/− 45°/0°/+ 45°/− 45°]

 Stringers

[0°/90°]2S

Ring

 Height (mm)

40

 Lay-up

[0°/90°]2S

Table 2

Lamina material properties

Parameter

Value

Young’s modulus, E11 (MPa)

57,765

Young’s modulus, E22 (MPa)

53,686

Shear modulus, G12 (MPa)

3065

Poisson’s ratio, ν12

0.048

Density, ρ (kg/m3)

1510

Ply thickness (mm)

0.33

3.2 Finite element model

Following experimental data described in previous section, a finite element model of cylinder was developed to simulate buckling and postbuckling process using ABAQUS software. Model contains about 18000 S4R-type shell elements with mean size 10 mm. Both ends of the cylinder were rigidly fixed to ensure actual gage length of 540 mm between reinforcing rings except the vertical degree of freedom for loaded end. Due to convergence problems and low loading rate, a quasi-static implicit solver for dynamic analysis was selected. As initial step of the study, a linear elastic constitutive model is used with properties given in Table 2. Then, nonlinear elastic model presented in Sect. 2 is implemented via UMAT subroutine.

4 Results and discussion

Figure 1 shows the experimental loading diagram for compression, a diagram obtained using linear elastic model, and a diagram obtained using constitutive relations, proposed in Sect. 2 of this paper. Coefficients presented in Table 3 for proposed model are used. Shear behavior is modelled with \( \alpha = 10^{ - 8} \) (MPa−3). The values from Table 3 are not uniquely obtained due to the lack of experimental data for their matching. But these values are in compatibility with available unidirectional data of Table 2 and give a good agreement with experimental results for compression shown in Fig. 1.
Fig. 1

Compressive load versus displacement diagrams

Table 3

Values of coefficients for constitutive relations Eq. 6 (1/MPa)

\( a_{11}^{0} \)

\( a_{22}^{0} \)

\( c_{11}^{{}} \)

\( c_{22}^{{}} \)

\( a_{12}^{0} \)

\( c_{12}^{0} \)

3.85E−03

3.85E−03

− 4.4E−03

− 4.4E−03

− 2E−3

2E−3

A possible reason of discrepancy between linear model prediction and experimental diagram from the very beginning of loading is that apparently only tension moduli from Table 2 are available, while compression moduli should be used for more accurate buckling analysis. There is no information about compression moduli in [5]. Moreover, compression moduli on the base of standard tests with the use of specimens of higher thickness than the ones in considered thin-walled shell may be different.

According to the experimental observation, it was no any shell damage detected until sudden collapse due to stringer buckling, so for this particular problem damage is neglected.

Deflections of the cylinder shell in normal direction to surface due to buckling presented on Fig. 2 for the experimental load cell displacement close to failure (3 mm). One can see that the two elastic models have different buckling shapes. Result with the proposed nonlinear anisotropic elastic model susceptible to stress state has the same form of buckling deflections as experimental measurements (Fig. 2c) [5].
Fig. 2

Shell displacements in radial direction due to buckling: a experimental measurements of displacements in one sector of the shell: 12.7 mm invard and 9.3 mm outward [5], b corresponding buckling zones, c similar buckling shapes obtained in simulation

5 Conclusion

The implementation of anisotropic elastic material model susceptible to the stress state and nonlinear shear within FEM software shows a good correlation of theoretical prediction with experimental results in tests of buckling of composite shells. Both loading diagrams and buckling shapes are close to the recorded ones during the composite shell compression test. Development and implementation of proposed model, including damage consideration, look as an effective tool for engineering applications.

Notes

Acknowledgements

This research was supported by the Russian Foundation for Basic Research (Grant no. 18-31-20026).

References

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Copyright information

© Shanghai Jiao Tong University 2018

Authors and Affiliations

  1. 1.Department of Aircraft EngineeringMoscow Aviation Institute (National Research University)MoscowRussian Federation

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