Structure of nonlinear elastic relationships for the highly anisotropic layer of a nonthin shell

Abstract

The problems of determining the constitutive (physical) relations for nonlinearly elastic composite materials (CM) were considered in numerous studies (see, for example, [ 1--4]). As a rule, the structural approach is used, i.e., most frequently, the problem of determining the effective characteristics of CM is solved by the same method as in the linearly elastic case. For example, the asymptotic method of averaging for the composites of regular structure was extended to the case of nonlinearly elastic media in [3,5]. In [6], a general method for establishing the constitutive relations for a CM reinforced with high-modulus fillers (fibers) was given on the basis of the asymptotic method using a great parameter--the ratio between the reinforcement and matrix stiffnesses. In [7], this method was applied to the case of nonlinearly elastic components. The physical relations for granular and fibrous CM with an elastic reinforcement and nonlinearly elastic matrix (the nonlinearity was assumed to be a square function of the components of the strain tensor) were obtained in [8,9] by averaging methods. Composites whose reinforcing components were also nonlinearly elastic were considered, in particular, in [10].The phenomenological approach is rarely applied to the problem of establishing the relation between the static and kinematic characteristics of CM (see, for example, [4, 11,12]). One of the most widely used ways for specifying the law of nonlinearly elastic deformation is expansion of the elastic potential in a tensorial series [11]. However, this approach requires a great number of terms in the series for a satisfactory description of the experimental data in a wide range of strains. For example, in [ 13], the experimental results for CFRP beyond the range of linear strain diagram were given. A comparison of the results obtained from a polynomial elastic potential with the experimental data shows that such an approach, in some cases, leads to great distinctions, both quantitative and qualitative, between the predicted and actual strain diagrams.In [14][15][16], an asymptotic method for analyzing the constitutive relations was developed, their classification depending on the properties of CM was carried out, and different types of such relations containing functions of small number of argumants (one or two) were obtained.In the present study, this method is used to construct simplified models of deformation for a transversely isotropic CM for cases different from those considered in [16]. These models are then used to formulate the physical relations for the material of a shell calculated by one of the shear theories.1. Let us consider a fibrous (unidirectionally reinforced) material which, due to irregular and arbitrary arrangement of fibers, can be regarded as transversely isotropic in the cross sections perpendicular to the direction ofreinforeement. Let the Ox ~ axis Kazan State Architectural Building Academy, Tatarstan, Russia.

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