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Simple constitutive models to represent the effect of mechanical damage and abrasion on the short-term load-strain response of geosynthetics
Published in António S. Cardoso, José L. Borges, Pedro A. Costa, António T. Gomes, José C. Marques, Castorina S. Vieira, Numerical Methods in Geotechnical Engineering IX, 2018
Equation 1 represents a generic polynomial model, where T is the load per unit width, ∊ is the axial tensile strain, ai is the polynomial coefficient of order i and n is the order of the polynomial. Equation 2 represents the tangent stiffness (Jt∊%) of the geosynthetic for a strain ∊ (in%), obtained by derivation of Equation 1. () T=∑i=0naiεi () Jtε%=dTdε=∑i=1naiεi−1
Simple constitutive models to represent the effect of mechanical damage and abrasion on the short-term load-strain response of geosynthetics
Published in António S. Cardoso, José L. Borges, Pedro A. Costa, António T. Gomes, José C. Marques, Castorina S. Vieira, Numerical Methods in Geotechnical Engineering IX, 2018
Equation 1 represents a generic polynomial model, where T is the load per unit width, ε is the axial tensile strain, ai is the polynomial coefficient of order i and n is the order of the polynomial. Equation 2 represents the tangent stiffness (Jtε%) of the geosynthetic for a strain ε (in%), obtained by derivation of Equation 1. () T=∑i=0naiεi () Jtε%=dTdε=∑i=1naiεi−1
Number Theory and Cryptographic Hardness Assumptions
Published in Jonathan Katz, Yehuda Lindell, Introduction to Modern Cryptography, 2020
We let G denote a generic, polynomial-time, group-generation algorithm. This is an algorithm that, on input 1n, outputs a description of a cyclic group G, its order q (with ║q║ = n), and a generator g∈G. The description of a cyclic group specifies how elements of the group are represented as bit-strings; we assume that each group element is represented by a unique bit-string. We require that there are efficient algorithms (namely, algorithms running in time polynomial in n) for testing whether a given bit-string represents an element of G, as well as for computing the group operation. This implies efficient algorithms for exponentiation in G (see Appendix B.2.3), computing inverses (the inverse of g is gq−1) and for sampling a uniform element h∈G (simply choose uniform x ∈ ℤq and set h:= gx). As discussed at the end of the previous section, although all cyclic groups of a given order are isomorphic, the representation of the group determines the computational complexity of mathematical operations in that group.
Unified three-dimensional finite elements for large strain analysis of compressible and nearly incompressible solids
Published in Mechanics of Advanced Materials and Structures, 2023
A. Pagani, P. Chiaia, M. Filippi, M. Cinefra
Numerous mechanical and aeronautical engineering applications, soft tissue modeling, and bio-inspired material deal with hyperelastic materials. Soft materials are used for the development of smart sensors, which can in turn be used for several applications, including structural health monitoring [1]. Hyperelastic materials are widely involved in muscular and cardiac tissue modeling [2, 3] and mechanical applications like bearings fabrications [4]. Over the years, many efficient models based on strain energy functions have been proposed for the simulation of hyperelastic media, each of them based on different phenomenological assumptions. Gent [5] proposed a model of rubber elasticity starting from the assumption of limiting chain extensibility. Khajehsaeid et al. [6] proposed a model based on a material molecular network. Famous and widely spread for their simplicity neo-Hookean and Mooney-Rivlin models for rubber-like materials are derived from the Ogden model [7], based on a generic polynomial expansion of principal stretches to describe the strongly nonlinear behavior of soft tissues. The common feature between these different models is the ability to describe strongly nonlinear behavior characterized by large strains and rotations. Many hyperelastic constitutive laws are derived assuming the decomposition of deformation gradient in a purely volumetric part and a purely isochoric part, decoupling the pure dilatation components from volume preserving terms [8]. The adoption of these models is justified by the analytical expression of the constitutive law that can be derived straightforwardly, but a more general derivation of constitutive law independent from the strain energy function adopted requires much more mathematical effort.
On continuous selections of polynomial functions
Published in Optimization, 2022
Feng Guo, Liguo Jiao, Do Sang Kim
Now we show that for generic CSP functions, the constant in Theorems 5.1– 5.3 can be replaced by 1/2. The next result can be seen as a generalization of the well known fact that the Łojasiewicz exponent for a generic polynomial (which is Morse in fact) is equal to 1/2.