# The development of a composite laminate macromodel for the analysis of stress-strain behavior in irregular zones of typical airframe

### Mathematics. Physics. Mechanics

### Аuthors

^{*},

^{**}

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

*e-mail: grischenko1911@gmail.com

**e-mail: design101@mai.ru

### Abstract

Preliminary assessment of stress-strain behavior is an important stage in design of any aircraft structure. Such assessment allows determining the first approximation of geometrical parameters of the designed structure elements already at the stage of its development. The more accurate the methods of stress-strain behavior assessment are, the lower are the costs of the subsequent design adjustments, which are implemented on the basis of the structure test results.Design calculations of composite laminate structures are carried out by mainly using composite laminate macromodels, which are based on the theory of laminated anisotropic material, and various finite element models. The main role of the finite element models in this calculation consists in determining the effective stress. To obtain the solution in the first case an assumption is made that the deformations are equal and constant along the packet thickness. In the second case it is possible to construct a composite laminate model, which would take into account the interlayer and interfacial interactions between the composite laminate components. However, in this case the required computing capacity grows manifold directly proportional to the number of layers.

The goal of this research is to develop the method of analysis of stress strain behavior of any composite laminate packet in irregular zone of typical airframe structures. This is attained by creating a special mathematical macromodel for numerical analysis of the deformation of an arbitrary laminate composite packet, which allows for possible movement of the layers relative to each other with taking into account the shear stiffness of the binder.

Only two-dimensional stress state is considered in this study. An assumption is made that the interlayer space in a composite laminate packet is filled exclusively with the binder, the mechanical properties of which are isotropic. In the framework of the developed model a multilayer laminate composite packet, which actually consists of

*N*monolayers with

*h*thickness, is considered to have

_{M}*N*orthotropic lamina with

*h*thickness and

_{K}*N-1*binder layers with

*h*thickness. Each orthotropic laminae in the model is considered as a composite monolayer with a somewhat higher percentage of fiber content. The binder in the model is considered to have isotropic mechanical properties.

_{C}Thus it is assumed that the packet consists of layers of 2 different materials. The interfacial connections are considered ideal, i.e. the deformations are constant along the materials interface. It is also assumed that composite monolayer can only be in two-dimensional stress state, while intermediate binder layer can be in three-dimensional stress state with the exception of longitudinal deformations along the Z axis. Therefore, the possibility of occurrence of shear deformations

*γ*and

_{XZ}*γ*and, subsequently, shear stresses τ

_{YZ}

_{XZ}_{ }and τ

*in the intermediate binder layer is not excluded from consideration.*

_{YZ.}The problem is reduced to the determination of the strain and deformation of composite layers according to the laws of the classical theory of elasticity of a laminate anisotropic material. For intermediate binder layers the deformations (including shear deformations) are determined first, the tangential shear, normal and equivalent strains are determined afterwards.

Interlayer shear deformations are defined by solving the equations of strain compatibility for the intermediate binder layer. This is done by using the dependence of the longitudinal deformation within the binder layer on the coordinate along the Z axis (boundary conditions are assumed as equal to zero):

where ε

_{l}

^{(2)}and ε

_{l}

^{(2)}are the deformations of the adjacent layers 1 and 2.

A calculation of a hypothetical loading case was carried out to investigate the capabilities of the methodology and analyze the results. A 40-layer composite laminate packet was considered. It was assumed that a certain load is applied to it. The load acts on a certain number of the upper packet layers (loaded layers).

The dependence of the layer and interlayer deformations on the coordinate along the packet thickness shows a strongly pronounced inverse proportionality. Interlayer shear deformations tend towards zero with greater intensity. Such dependence is most pronounced for the packet, which is reinforced in only one direction.

The shape of deformation distribution in the layers is independent from the amount of the applied load, the loading condition and calculation pattern. This means that there is a certain invariant characteristic of the deformation of the composite laminate packet, which depends only on the structure and properties of the packet. This allows assessing the stress-strain behavior of such packet at the stage when the load itself and loading condition are unknown.

It is necessary to gradually increase the thickness in the irregular zone to enable a more uniform loading of the composite laminate layers. However, based on the obtained solutions for the interlayer deformation it is possible to conclude that the interlayer shift directly depends on the size of the transition zone. Therefore the problem of determining the optimal size of the transition zone emerges. The optimal zone should allow gradual loading of the layers without exposing the interlayer binder to dangerous strains.

If there are shear stresses between the layers, they usually substantially prevail over normal ones. Thus delamination calculation can be based on the shear strength of the binder material. However, creation of an integrated strength criterion may be required.

There is a significant increase of the interlayer deformations in areas where the load is transferred from the layer with reinforcement angle 90° to the layer with reinforcement angle 0°. This is caused by the fact that the difference in stiffness of these layers is too big. When the load is transferred from the layer with reinforcement angle 0° to the layer with reinforcement angle 90°, a reversed interlayer effect can be observed, which consists in small values of the interlayer deformations and strains.

### Keywords:

composite material, stress-strain behavior, irregular zones, interlaminar shift### References

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