Alternate Stress and Conjugate Strain Measures, and Mixed Variational Formulations Involving Rigid Rotations, for Computational Analyses of Finitely Deformed Solids, with Application to Plates and Shells-i

Attention is focused in this paper on: (i) definitions of alternate measures of “stress-resultants” and “stress-couples” in a finitely deformed shell (finite mid-plane stretches as well as finite rotations); (ii) mixed variational principles for shells, undergoing large mid-plane stretches and large rotations, in terms of a stress function vector and the rotation tensor. In doing so, both types of polar decomposition, namely rotation followed by stretch, as well as stretch followed by rotation, of the shell midsurface, are considered; (iii) two alternate bending strain measures which depend on rotation alone for a finitely deformed shell; (iv) objectivity of constitutive relations, in terms of these alternate strain/“stress-resultants”, and “stress-couple” measures, for finitely deformed shells. To motivate these topics, and for added clarity, a discussion of relevant alternate stress measures, work-conjugate strain measures, and mixed variational principles with rotations as variables, is presented first in the context of three-dimensional continuum mechanics. Comments are also made on the use of the presently developed theories in conjunction with mixed-hydrid finite element methods. Discussion of numerical schemes and results is deferred to the Part II of the paper, however. NOMENCLATURE undeformed body deformed body curvilinear coordinates in B, and convected coordinates in b another set of curvilinear coordinates in b under a symbol denotes a vector under a symbol denotes a second-order tensor A&? denotes a second-order tensor in dyadic notation AsBig, C’B: gradient operator base vectors in B convected base vectors in b another set of base vectors in b deformation gradient tensor of a 3-D continuum polar decomposition of E stretch tensors rigid rotation tensor unit normal to an oriented surface in B unit normal to an oriented surface in b absolute determinant of E 8.8 IJ=.n Cauchy stress tensor Kirchhoff stress tensor first Piola-Kirchhoff stress tensor Yl’his paper is presented to my good friend, Prof. Kyuichiro Washizu on the occasion of his 60th birthday and the completion of a distinguished academic career at the University of Tokyo. SRegents’ Professor of Mechanics. second Piola-Kirchhoff stress tensor Biot or Lure’ stress tensor symmetrized Biot or Lure’ stress tensor “convected stress tensor” two other “induced” stress tensors RT.7,R &+.zj= 1/2(T* t T*=) = &r,BT Eulerean strain rate tensor velocity gradient tensor right-Cauchy-Green deformation tensor Ieft-Cauchy-Green deformation tensor Green-Lagrange strain tensor Almansi strain tensor natural logarithm rate of increase of internal energy (“stressworking-rate”) per unit undeformed volume strain energy per unit initial volume “complementary energy” undeformed mid-surface of a shell deformed mid-surface of a shell base vectors on S unit normal to S second fundamental form of S base vectors on s unit normal to s mid-surface deformation gradient polar decomposition of & mid-plane in a Kirchoff-Love theory BAe BT.$ Cauchy stress-resultant tensor for a shell Cauchy stress-couple tensor for a shell first Piola-Kirchhoff stress resultant tensor second Piola-Kirchhoff stress resultant tensor Biot-Lure’ stress-resultant tensor