The Nonlinear Field Theories in Mechanics

Abstract

We can now proceed in precisely the same manner as in section 2 to establish convergence of the iteration scheme.If ro = 0, the operator L[u] as defined above is not coercive and lemma 3.1 does not hold. In this case, we adopt a different procedure. Let i be any positive real number. We apply the operator (u . ' 17) + 2 to (3.3) and obtainWe now seta coercive operator in P(Q) and L[u]-l exists. We can now apply the operator L[u]-l to (3.8), define p = L[u]-l x x ((u * V) + 1) p and set up an iteration scheme in an analogous fashion as before. References 1 M. J. CROCHET and K. WALTERS, Numerical methods in non-Newtonian fluid mechanics, Ann. Rev. Fluid Mech. 15 (1983), 241 -260. 2 H. GIESEKUS, A unified approach to a variety of constitutive models for polymer fluids based on the concept of configuration dependent molecular mobility, Rheol. Acta 21 (19SZ), 386-3375. 3 M. S. GREEN and A. V. TOBOLSRY, A new approach to the theory of relaxing polymeric media, J. Chem. Phys. 14 (1946), 80-100. 4 T. KATO, Perturbation Theory for Linear Operators, Springer, 1966. 5 0. A. LADY~ENSRAYA, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, 1969. 6 A. I. LEONOV, Nonequilihrium thermodynamics and rheology of viscoelastic polymer media, Rheol. Acta 16 (1976), 85-98. 7 A. 5. LODGE, A network theory of flow birefringence and stress in concentrated polymer solutions, Trans. Faraday Sac. 62 (1956), 120-130. 8 111. NIGGEMANN, A model equation for non-Newtonian fluids, Math. Meth. Appl. Sci. S (1981), 200-217. 9 J. G. OLDROYD, Non-Newtonian effects in steady mation of some idealized elasticoviscous liquids, Proc.

Physics-based modeling of brittle fracture: cohesive formulations and the application of meshfree methods

“…The classical condition for strain localization states that the acoustical tensor q(N) is singular for a particular choice of N, which then defines the normal to the surface across which a discontinuity appears. This result can be derived by admitting a deformation field containing a jump in the velocity gradient as with acceleration waves [40,88], or by admitting a jump in the displacement field [3]. Though the details of the analyses differ, the result in these, and many other examples, is the same: the loss of positive definiteness of the acoustical tensor marks the bifurcation of the solution from a smooth field to a solution admitting discontinuous features.…”

“…Although the analysis of softening due to a network of cohesive surfaces in Section 2.1.3 or theoretical conditions for the appearance of discontinuous bifurcations [40,88] may suggest an adaptive approach is needed for modeling cohesive fracture, the implementation of a simulation procedure introduces a number of parameters not prescribed by theory. The parameters we introduce in our moving tip model are shown in Figure 3.7 (a).…”

Aspects of the Phenomenological Theory of Elastic-Plastic Deformation

“…That the strain energy should depend on H via C is necessary and sufficient for the symmetry of the Cauchy stress T, related to P by P = TF * (see [1,10]). This, of course, is equivalent to (4) 2 .…”

An anisotropic viscoelastic fibre–matrix model at finite strains: Continuum formulation and computational aspects

“…Similarly, the part ψ neq M of the free energy function depends on the invariants of C e M which are identical to those of the corresponding elastically relaxing left Cauchy-Green tensor b e M = F e M F e T M . Then, from the well-known result of isotropic function theory, see for example [7,16,33], the corresponding equilibrium and non-equilibrium Kirchhoff stress tensors τ ∞ M = FS ∞ M F T and τ neq M = FS neq M F T can be equivalently given by,…”