PSI - Issue 13

Evgeny Lomakin et al. / Procedia Structural Integrity 13 (2018) 664–669 Evgeny Lomakin and Boris Fedulov / Structural Integrity Procedia 00 (2018) 000–000

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1. Introduction Composite materials became an industrial standard in lightweight structures. Stress analysis of structures of such kind of materials is a usual practice for modern engineers. Nevertheless, there is no generally accepted solution of this problem for reliable modelling of first stage of composite material deformation with elastic response. No one popular engineering software package as Abaqus, Ansys, or Nastran has any built-in material model, which can take into account such effects as different stiffness in cases of compression and tension or stiffness drop in the case of plane shear loading. Underestimating of these effects as a result gives wrong predictions of stress distribution in general and leads to wrong analysis of static and fatigue failure. In practice, there are many examples of essential discrepancy in the cases with bending loads such as deflections of composite wing, wrong buckling load predictions, etc. This research deals with this problem and performs a new elastic material model for composite materials, which is susceptible to the stress state type, and has an ability to capture a shear stiffness reduction during plane shear loading. 2. Formalization of stress state For developing of mathematical model taking into account the effect of different possible response of composite materials to the type of loading, the formalization of stress state is required. By means of = /3 which is the hydrostatic stress component, and 0 σ (3/2)S S ij ij  , which is the equivalent stress, where = − is a deviator stress, it is possible to formulate the parameter = / 0 . This parameter is a good candidate for proposed formalization due to its mechanical sense, invariant nature and scalar simplicity. This introduced parameter can be found in the literature under the name of the stress triaxiality. 3. Formalization of shear parameter The parameter which can describe the degree of shear loading, for example under plane stress conditions, can be formalized as a shear stress component in the case of plane shear loading and formulated in invariant form as , ij ij Q D   where 0 1/ 2 0 1/ 2 0 0 . 0 0 0 ij D            This obviously gives in basic coordinate system 12 . Q   4. Constitutive equations It is possible to show that introducing parameters of stress state Q into coefficients of elastic potential   Φ , , ijkl ij kl A Q     gives a robust set of constitutive equations that can capture all effects of elastic nonlinearities exhibited by composite materials. In case of plane stress problem, the constitutive equations have the following form:     2 11 1111 11 1122 22 11 1 0 1 3 3 Φ , 3 2 2 A A                            

  

 

  

 

1 3 3 2 

3 2



2

 

 





 

22 1 0        Φ



 

 

A

A

,

22

1122

11 2222

22