Simple extension

Abstract: It is shown in this part how the theory of large elastic deformations of incompressible isotropic materials, developed in previous parts, can be used to interpret the load-deformation curves obtained for certain simple types of deformation of vulcanized rubber test-pieces in terms of a single stored-energy function. The types of experiment described are: (i) the pure homogeneous deformation of a thin sheet of rubber in which the deformation is varied in such a manner that one of the invariants of the strain, I 1 or I 2 , is maintained constant; (ii) pure shear of a thin sheet of rubber (i.e. pure homogeneous deformation in which one of the extension ratios in the plane of the sheet is maintained at unity, while the other is varied); (iii) simultaneous simple extension and pure shear of a thin sheet (i.e. pure homogeneous deformation in which one of the extension ratios in the plane of the sheet is maintained constant at a value less than unity, while the other is varied); (iv) simple extension of a strip of rubber; (v) simple compression (i.e. simple extension in which the extension ratio is less than unity); (vi) simple torsion of a right-circular cylinder; (vii) superposed axial extension and torsion of a right-circular cylindrical rod. It is shown that the load-deformation curves in all these cases can be interpreted on the basis of the theory in terms of a stored-energy function W which is such that δ W /δ I 1 is independent of I 1 and I 2 and the ratio (δ W /δ I 2 ) (δ W /δ I 1 ) is independent of I 1 and falls, as I 2 increases, from about 0*25 at I 2 = 3.

Abstract: We introduce two new tools that can be useful in nonlinear observer and output feedback design. The first one is a simple extension of the notion of homogeneous approximation to make it valid both at the origin and at infinity (homogeneity in the bi-limit). Exploiting this extension, we give several results concerning stability and robustness for a homogeneous in the bi-limit vector field. The second tool is a new recursive observer design procedure for a chain of integrator. Combining these two tools, we propose a new global asymptotic stabilization result by output feedback for feedback and feedforward systems.

Abstract: We propose a simple extension of the well-known Riemann solver of Osher and Solomon (Math Comput 38:339---374, 1982) to a certain class of hyperbolic systems in non-conservative form, in particular to shallow-water-type and multi-phase flow models To this end we apply the formalism of path-conservative schemes introduced by Pares (SIAM J Numer Anal 44:300---321, 2006) and Castro et al (Math Comput 75:1103---1134, 2006) For the sake of generality and simplicity, we suggest to compute the inherent path integral numerically using a Gaussian quadrature rule of sufficient accuracy Published path-conservative schemes to date are based on either the Roe upwind method or on centered approaches In comparison to these, the proposed new path-conservative Osher-type scheme has several advantages First, it does not need an entropy fix, in contrast to Roe-type path-conservative schemes Second, our proposed non-conservative Osher scheme is very simple to implement and nonetheless constitutes a complete Riemann solver in the sense that it attributes a different numerical viscosity to each characteristic field present in the relevant Riemann problem; this is in contrast to centered methods or incomplete Riemann solvers that usually neglect intermediate characteristic fields, hence leading to excessive numerical diffusion Finally, the interface jump term is differentiable with respect to its arguments, which is useful for steady-state computations in implicit schemes We also indicate how to extend the method to general unstructured meshes in multiple space dimensions We show applications of the first order version of the proposed path-conservative Osher-type scheme to the shallow water equations with variable bottom topography and to the two-fluid debris flow model of Pitman & Le Then, we apply the higher-order multi-dimensional version of the method to the Baer---Nunziato model of compressible multi-phase flow We also clearly emphasize the limitations of our approach in a special chapter at the end of this article

Abstract: A plot of principal stress difference versus principal extension ratios has been used as a graphic representation of general deformation. Two analytic forms of the strain energy function for isotropic, incompressible materials are suggested. These involve five or nine terms, the coefficients of which are found by regression to the general deformation plot. The resulting stress–strain equations are used to predict particular deformations, for example, simple extension, and are also evaluated in model engineering desing experiments. These experiments use interative techniques to predict the shapes and pressures of inflated diaphragms and tubes, and it is shown that the equations lead to accurate results even at relatively high extensions.