International audienceFor laminated plates, the displacement field can be approximated in each layer by a third-order Taylor--Young expansion in thickness. These involve (sections 3.1 and 3.2) that the highest order term of transverse shear is of first-order in thickness. Then we are motivated to consider a simplified theory based on the thickness-wise expansion of the potential energy truncated at third order in thickness. The equilibrium equations imply local constraints on the through-thickness derivatives of the zero-order displacement field in each layer. These lead to an analytical expression for two-dimensional potential energy in terms of the zero-order displacement field and its derivatives that includes non-standard transverse shearing energy and coupled bending--stretching energy. As a consequence this potential energy satisfies the stability condition of Legendre--Hadamard which is necessary for the existence of a minimizer
An asymptotic reduction method is introduced to construct a rod theory for a linearized general anisotropic elastic material for space deformation. The starting point is Taylor expansions about the central line in rectangular coordinates, and the goal is to eliminate the two cross-section spatial variables in order to obtain a closed system for displacement coefficients. This is first achieved, in an ‘asymptotically inconsistent’ way, by deducing the relations between stress coefficients from a Fourier series for the lateral traction condition and the three-dimensional (3D) field equation in a pointwise manner. The closed system consists of 10 vector unknowns, and further refinements through elaborated calculations are performed to extract bending and torsion terms and to obtain recursive relations for the first- and second-order displacement coefficients. Eventually, a system of four asymptotically consistent rod equations for four unknowns (the three components of the central-line displacement and the twist angle) are obtained. Six physically meaningful boundary conditions at each edge are obtained from the edge term in the 3D virtual work principle, and a one-dimensional rod virtual work principle is also deduced from the weak forms of the rod equations.
International audienceWe approximate the displacement field in a shell by a fifth-order Taylor–Young expansion in thickness. The model is derived from the truncation of the potential energy at fifth order. The equilibrium equations imply local constraints on the through-thickness derivatives of the zero-order displacement field. This leads to an analytical expression for the two-dimensional potential energy of a shell in terms of the zero-order displacement field and its derivatives that include non-standard transverse shearing and normal stress energy. Then we derive the equation of equilibrium and the boundary conditions
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