Nonlinear Fractional Derivative Stress-Strain Relations for Polymer Gels Based on the Generalized Maxwell Model

2009 ◽  
Author(s):  
Masataka Fukunaga ◽  
Nobuyuki Shimizu
Keyword(s):  
Fractional Derivative ◽  
Memory Effect ◽  
Basic Assumption ◽  
Dynamical Behavior ◽  
Maxwell Model ◽  
Polymer Gels ◽  
Stress Strain ◽  
Stress Strain Relation ◽  
Nonlinear Stress ◽  
Generalized Maxwell Model

In this paper, we formulate two nonlinear stress-strain relations including memory effect in the dynamical behavior of gels that are the kind of viscoelastic materials. The basic assumption of the model is made that the gels consist of blobs of high polymers. Hereditary response of blobs to the stress determines the average stress-strain relation of the material. Two stress-strain relations are derived for different models of gels. These stress-strain relations are compared with the fractional derivative version of Lodge’s rubber-like liquids and the empirical nonlinear fractional derivative model proposed by Fukunaga et. al. at FDA08.

Parametric Variational Principle for Bi-Modulus Materials and Its Application to Nacreous Bio-Composites

2016 ◽  
Vol 08 (06) ◽  
pp. 1650082 ◽  
Author(s):  
Liang Zhang ◽  
Huiting Zhang ◽  
Jian Wu ◽  
Bo Yan ◽  
Mengkai Lu
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Variational Principle ◽  
Principal Stress ◽  
Stress Strain ◽  
Element Analysis ◽  
Parametric Variational Principle ◽  
Strain Relation ◽  
Stress Strain Relation ◽  
Nonlinear Stress

Bi-modulus materials have different moduli in tension and compression and the stress–strain relation depends on principal stress that is unknown before displacement is determined. Establishment of variational principle is important for mechanical analysis of materials. First, parametric variational principle (PVP) is proposed for static analysis of bi-modulus materials and structures. A parametric variable indicating state of principal stress is included in the potential energy formulation and the nonlinear stress–strain relation is evolved into a linear complementarity constraint. Convergence of finite element analysis is thus improved. Then the proposed variational principle is extended to a dynamic problem and the dynamic equation can be derived based on Hamilton’s principle. Finite element analysis of nacreous bio-composites is performed, in which a unilateral contact behavior between two hard mineral bricks is modeled using the bi-modulus stress–strain relation. Effective modulus of composites can be determined numerically and stress mechanism of “tension–shear chain” in nacre is revealed. A delayed effect on stress propagation is found around the “gaps” between mineral bricks, when a tension force is loaded to nacreous bio-composites dynamically.

Nonlinear Fractional Derivative Models of Viscoelastic Impact Dynamics Based on Entropy Elasticity and Generalized Maxwell Law

2010 ◽  
Vol 6 (2) ◽  
Author(s):  
Masataka Fukunaga ◽  
Nobuyuki Shimizu
Keyword(s):  
Fractional Derivative ◽  
Basic Assumption ◽  
Dynamical Behavior ◽  
Impact Dynamics ◽  
Viscoelastic Materials ◽  
Free Response ◽  
Scale Free ◽  
Collective Response ◽  
Elementary Constituent ◽  
Entropy Elasticity

Two types of models are proposed for describing nonlinear fractional derivative dynamical behavior of viscoelastic materials subject to impulse forces. The models are derived based on the thermodynamic elasticity in terms of entropy and on the “scale-free response of the material” under the basic assumption that the viscoelastic materials consist of stable coils of polymers, which we refer to as blobs. The blobs, which may be connected to each other by chemical bonds or physical bonds, are considered here as the elementary constituent of viscoelastic materials from which the nonlinear fractional derivative models are derived. Responses of individual blobs can determine the net collective response of the viscoelastic material to impulse forces. From the above consideration, two types of models are proposed in which the force elements or the stress elements are connected by the generalized Maxwell law.

Effects of Nonlinear Stress-Strain Rate Relation on Deformation and Fracture of Materials in Creep Range

1979 ◽  
Vol 101 (4) ◽  
pp. 369-373 ◽  
Author(s):  
Ryuichi Ohtani ◽  
Shuji Taira
Keyword(s):  
High Temperature ◽  
Industrial Development ◽  
Practical Investigations ◽  
High Temperature Strength ◽  
Stress Strain ◽  
Stress Strain Relation ◽  
Deformation And Fracture ◽  
Nonlinear Stress ◽  
Fracture Of Materials ◽  
Rate Relation

Fundamental and practical investigations on high-temperature strength of materials have made significant contributions to industrial development for the last twenty-five years. However, an additional effort is required to make clear the effects of dominant factors on the deformation and fracture of materials, since the knowledge obtained to date are not enough to explain the overall phenomena of high temperature strength. In this paper the importance of the effects of nonlinearlity and time-dependence of stress-strain relation on the well-known creep behaviors are reviewed and reexamined on the basis of our laboratory study conducted over the last twenty years.

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