Advances in the Mechanics of Plates and Shells: The Avinoam by D. Durban, Dan Givoli, J.G. Simmonds

By D. Durban, Dan Givoli, J.G. Simmonds

The optimum keep watch over of versatile constructions is an lively sector of study. the most physique of labor during this region is anxious with the keep an eye on of time-dependent displacements and stresses, and assumes linear elastic stipulations, specifically linear elastic fabric habit and small defor- tion. See, e. g. , [1]–[3], the collections of papers [4, 5], and references therein. however, within the current paper we think about the static optimum regulate of a constitution made from a nonlinear elastic fabric and und- going huge deformation. an immense software is the suppression of static or quasi-static elastic deformation in versatile house constructions comparable to components of satellites by way of regulate so much [6]. sunlight rad- tion and radiation from different assets set off a temperature box within the constitution, which in flip generates an elastic displacement box. The displacements needs to frequently fulfill definite obstacles dictated through the allowed operating stipulations of varied orientation-sensitive tools and antennas within the area motor vehicle. for instance, a parabolic reflector may possibly stop to be powerful while present process huge deflection. The elastic deformation should be diminished by way of use of keep an eye on so much, that may be imp- mented through mechanically-based actuators or extra sleek piezoelectric units. whilst the constitution into account is made up of a rubb- like fabric and is present process huge deformation, nonlinear fabric and geometric results has to be taken under consideration within the research.

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According to Atluri and Murakawa [10, 11], the inversion of 1st Piola-Kirchhoff stress tensor and the associated strain tensor is not one-to-one. The symmetric part of the Biot-Lure stress tensor or the Jaumann stress tensor and the conjugated strain tensor leads to a one-to-one mapping. (60) and (61), we obtain the expression for the mechanical power written as (62) (63) (64) (65) where z = z A 3 is the Lagrangian stretching vector along ξ 3. In contrast to the three-dimensional continuum mechanics, the mechanical power of the shell 27 consists of the two parts: one is associated with the expression for the conjugate relationships between stress and strain measures, and the other the expression for the AMB equation.

Therefore, the invariance condition of strain energy function is not satisfied when the assumptions (G1) and (G2) are used. (70). This fact shows the importance of using the correct assumptions for the stress states . 5. Variational Principle According to Atluri [1-4] and Suetake, Iura and Atluri [17], the generalized functional for the shell is given by (71) where denotes the Lagrange multiplier, the prescribed value at the boundary, and W0 and c 3 are given, under each assumption, by The variables subjected to variation in Eq.

The vector φ, however, does not exist so that the strain energy function can not be expressed in terms of φ. 3 OBJECTIVITY We consider the objectivity or the invariance of strain energy function. Since the present strain vectors cα, b α and z are associated with the reference base vectors A i , they are the Lagrangian measures. It is, therefore, clear that the present strain vectors are objective (see Ogden [12]). Let Q be a proper orthogonal tensor. The objective strain vector is transformed such that (67) where ( )* denotes the value after the rigid rotation.

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