A complete set of independent and physically meaningful invariants in the mechanics of solids reinforced by two families of fibres

It has recently been shown [2, 3] that only seven of the classical deformation invariants employed in hyperelasticity of solids reinforced by two families of unidirectional fibres are independent. This short communication demonstrates a manner in which such a set of seven invariants is conveniently identified without much deviation from well-known features that characterise their classical counterparts. It also shows that, unlike several of their classical counterparts, these newly identified invariants have all their own physical meaning. This new development is immediate applicable on mass-growth problems of tissue that preserve fibre direction [1] and, notably, on problems involving mass-growth of a circular tube reinforced by two families of helices wound symmetrically around the tube in opposite directions

Mass-growth of a finite tube reinforced by a pair of helical fibres

Several types of tube-like fibre-reinforced tissue, including the layers of arteries and veins, different kinds of muscle, biological tubes as well as plants and trees, are reinforced by a pair of helical fibres wound symmetrically around the tube axis in opposite directions. In many cases, this kind of biological structures grow in an axially symmetric manner that preserves their own shape as well as the direction and shape of their embedded pair of helical fibres. This study considers and investigates the influence that preservation of fibre direction exerts on pseudo-elastic (elastic-like) mass-growth modelling of the described fibre-reinforced structure. Complete sets of necessary conditions that enable the implied axisymmetric tube mass-growth to take place are sought, found and presented. These hold in addition to, and simultaneously with standard kinematic relations and equilibrium equations met in conventional hyperelasticity. They thus render this mass-growth mathematical model the properties of an apparently overdetermined boundary value problem. However, the additional information they provide leads to identification of admissible classes of strain energy densities for growth that enable realisation of the implied type of tube mass-growth. The analysis is applicable to several different types of mass-growth of tube-like tissue reinforced by a pair of symmetrically wound helical fibres. This is demonstrated with an application which considers that mass-growth of the fibre-reinforced tube takes place in an incompressible manner; namely, in a non-isochoric manner that along with fibre direction, it also preserves the material density of the growing tube

The generalized viscoelastic spring

A spring/rod model is presented that describes one-dimensional behaviour of solids susceptible to large or small viscoelastic deformation. Derivation of its constitutive equation is underpinned by the fact that the internal energy, which the elastic part of deformation stores into the spring, changes in time with the observed strain as well as with some, unknown part of the strain-rate. The latter emerges through the action of a viscous flow potential and is the source of inelastic deformation. Thus, unlike its conventional viscoelasticity counterparts, the model does not postulate a priori a rule that relates strain with viscous flow formation. Instead, it considers that such a rule, as well as other important features of combined elastic and inelastic material response, should become known a posteriori through the solution of a relevant, well-posed boundary value problem. This communication begins with considerations compatible with large viscoelastic deformations, and gradually progresses through simpler modelling situations. The latter also include the case of small viscoelastic strain that underpins formulation of classical, spring-dashpot viscoelastic models. In an example application, a relevant closed form solution is obtained for a spring undergoing small viscoelastic deformation under the influence of a viscous flow potential which is quadratic in the stress

Plane Strain Polar Elasticity Of Fibre-Reinforced Functionally Graded Materials and Structures

This study investigates the flexural response of a linearly elastic rectangular strip reinforced in a functionally graded manner by a single family of straight fibres resistant in bending. Fibre bending resistance is associated with the thickness of fibres which, in turn, is considered measurable through use of some intrinsic material length parameter involved in the definition of a corresponding elastic modulus. Solution of the relevant set of governing differential equations is achieved computationally, with the use of a well-established semi-analytical mathematical method. A connection of this solution with its homogeneous fibre-reinforced material counterpart enables the corresponding homogeneous fibrous composite to be regarded as a source of a set of equivalent functionally graded structures, each one of which is formed through inhomogeneous redistribution of the same volume of fibres within the same matrix material. A subsequent stress and couple-stress analysis provides details of the manner in which the flexural response of the polar structural component of interest is affected by certain types of inhomogeneous fibre distribution

On the constitutive modeling of dual-phase steels at finite strains: a generalized plasticity based approach

In this work we propose a general theoretic framework for the derivation of constitutive equations for dual-phase steels, undergoing continuum finite deformation. The proposed framework is based on the generalized plasticity theory and comprises the following three basic characteristics: 1.A multiplicative decomposition of the deformation gradient into elastic and plastic parts. 2.A hyperelastic constitutive equation 3.A general formulation of the theory which prescribes only the number and the nature of the internal variables, while it leaves their evolution laws unspecified. Due to this generality several different loading functions, flow rules and hardening laws can be analyzed within the proposed framework by leaving its basic structure essentially unaltered. As an application, a rather simple material model, which comprises a von-Mises loading function, an associative flow rule and a non-linear kinematic hardening law, is proposed. The ability of the model in simulating simplified representation of the experimentally observed behaviour is tested by two representative numerical examples

On the constitutive modeling of dual-phase steels at finite strains: a generalized plasticity based approach

In this work we propose a general theoretic framework for the derivation of constitutive equations for dual-phase steels, undergoing continuum finite deformation. The proposed framework is based on the generalized plasticity theory and comprises the following three basic characteristics: 1.A multiplicative decomposition of the deformation gradient into elastic and plastic parts. 2.A hyperelastic constitutive equation 3.A general formulation of the theory which prescribes only the number and the nature of the internal variables, while it leaves their evolution laws unspecified. Due to this generality several different loading functions, flow rules and hardening laws can be analyzed within the proposed framework by leaving its basic structure essentially unaltered. As an application, a rather simple material model, which comprises a von-Mises loading function, an associative flow rule and a non-linear kinematic hardening law, is proposed. The ability of the model in simulating simplified representation of the experimentally observed behaviour is tested by two representative numerical examples

On the characterisation of polar fibrous composites when fibres resist bending - PART II: connection with anisotropic polar linear elasticity

This continuation of Part I (Soldatos, 2018) aims to make a connection between the polar linear elasticity for fibre-reinforce materials due to (Spencer and Soldatos, 2007; Soldatos, 2014, 2015) with the anisotropic version and the principal postulates of its counterpart due to (Mindlin and Tiersten, 1963). The outlined analysis, comparison and discussions are purely theoretical, and aim to collect and classify valuable information regarding the nature of continuous as well as weak discontinuity solutions of relevant well-posed boundary value problems. Emphasis is given on the fact that the compared pair of theoretical models has a common theoretical background (Cosserat, 1909) but different kinds of origin. Some new concepts and features, introduced in Part I, in association with linear elastic behaviour of materials having embedded fibres resistant in bending, are thus shown relevant to more general linearly elastic, anisotropic, Cosserat-type material behaviour. The different routes followed for the origination of the compared pair of models is known to produce identical results in the case of conventional (non-polar) linear elasticity. The same is here found generally non true in the polar elasticity case, although considerable similarities are also observed. No definite answers are provided regarding the manner in which existing differences might be bridged or, if at all possible, eliminated. These are matters that require further study and thorough investigation