scholarly journals On the existence of elastic minimizers for initially stressed materials

2019 ◽  
Vol 377 (2144) ◽  
pp. 20180074 ◽  
Author(s):  
D. Riccobelli ◽  
A. Agosti ◽  
P. Ciarletta
Keyword(s):  
Stress Tensor ◽  
Initial Stress ◽  
Applied Mathematics ◽  
Deformation Gradient ◽  
Local Existence ◽  
Initial Stresses ◽  
Reference Configuration ◽  
Mathematical Framework ◽  
Tensor Function ◽  
Symmetric Tensors

A soft solid is said to be initially stressed if it is subjected to a state of internal stress in its unloaded reference configuration. In physical terms, its stored elastic energy may not vanish in the absence of an elastic deformation, being also dependent on the spatial distribution of the underlying material inhomogeneities. Developing a sound mathematical framework to model initially stressed solids in nonlinear elasticity is key for many applications in engineering and biology. This work investigates the links between the existence of elastic minimizers and the constitutive restrictions for initially stressed materials subjected to finite deformations. In particular, we consider a subclass of constitutive responses in which the strain energy density is taken as a scalar-valued function of both the deformation gradient and the initial stress tensor. The main advantage of this approach is that the initial stress tensor belongs to the group of divergence-free symmetric tensors satisfying the boundary conditions in any given reference configuration. However, it is still unclear which physical restrictions must be imposed for the well-posedness of this elastic problem. Assuming that the constitutive response depends on the choice of the reference configuration only through the initial stress tensor, under given conditions we prove the local existence of a relaxed state given by an implicit tensor function of the initial stress distribution. This tensor function is generally not unique, and can be transformed according to the symmetry group of the material at fixed initial stresses. These results allow one to extend Ball's existence theorem of elastic minimizers for the proposed constitutive choice of initially stressed materials. This article is part of the theme issue ‘Rivlin's legacy in continuum mechanics and applied mathematics’.

Fundamentals of generalized rigidity matrices for multi-layered media

1983 ◽  
Vol 73 (3) ◽  
pp. 749-763
Author(s):  
Maurice A. Biot
Keyword(s):  
Plane Strain ◽  
Initial Stress ◽  
Three Dimensional ◽  
Layered Media ◽  
Mathematical Structure ◽  
Irreversible Thermodynamics ◽  
Initial Stresses ◽  
General Context ◽  
Viscoelastic Media ◽  
Linear Irreversible Thermodynamics

abstract Rigidity matrices for multi-layered media are derived for isotropic and orthotropic layers by a simple direct procedure which brings to light their fundamental mathematical structure. The method was introduced many years ago by the author in the more general context of dynamics and stability of multi-layers under initial stress. Other earlier results are also briefly recalled such as the derivation of three-dimensional solutions from plane strain modes, the effect of initial stresses, gravity, and couple stresses for thinly laminated layers. The extension of the same mathematical structure and symmetry to viscoelastic media is valid as a consequence of fundamental principles in linear irreversible thermodynamics.

Initial Stress in Biomechanical Models of Atherosclerotic Plaques

2011 ◽  
Author(s):  
L. Speelman ◽  
A. C. Akyildiz ◽  
J. J. Wentzel ◽  
E. H. van Brummelen ◽  
J. Jukema ◽  
...  
Keyword(s):  
Blood Pressure ◽  
Finite Element ◽  
Initial Stress ◽  
Atherosclerotic Plaque ◽  
Initial Stresses ◽  
Atherosclerotic Plaques ◽  
Finite Element Models ◽  
Biomechanical Models ◽  
Fibrous Cap ◽  
Stress Studies

Rupture of the cap of an atherosclerotic plaque is instigated when the stresses in the cap due to the blood pressure exceed the local cap strength. Image based computational finite element models of atherosclerotic plaques are widely used to compute stresses in the fibrous cap. These models are often based on pressurized geometries. The shape of the plaque is determined by the blood pressure at the time of imaging, and thus contains initial stresses (IS) and strains, which are generally ignored in plaque stress studies.

A TWO-DIMENSIONAL STATIC DEFORMATION OF AN IRREGULAR INITIALLY STRESSED MEDIUM

2008 ◽  
Vol 22 (20) ◽  
pp. 3473-3485
Author(s):  
M. M. SELIM
Keyword(s):  
Closed Form ◽  
Initial Stress ◽  
Half Space ◽  
Static Deformation ◽  
Initial Stresses ◽  
Two Dimensional ◽  
Eigenvalue Approach ◽  
Line Load ◽  
Notable Effect ◽  
Approach Method

The paper discusses the problem of a two-dimensional static deformation as the result of normal line-load acting inside an irregular initially stressed isotropic half-space. The eigenvalue approach method has been used. The irregularity is expressed by a rectangle shape. Further, the results for the displacements and stresses have been derived in the closed form. The effect of initial stress and irregularity are shown graphically. It was found that the initial stresses as well as irregularity have a notable effect on this deformation.

Circumferential waves in pre-stressed functionally graded cylindrical curved plates

2014 ◽  
Vol 21 (1) ◽  
pp. 87-97 ◽  
Author(s):  
Jiangong Yu ◽  
Chuanzeng Zhang ◽  
Xiaoming Zhang
Keyword(s):  
Initial Stress ◽  
Wave Equations ◽  
Axial Direction ◽  
Variable Coefficients ◽  
Functionally Graded ◽  
Initial Stresses ◽  
Polynomial Method ◽  
Coupled Wave Equations ◽  
Graded Material ◽  
Polynomial Series

AbstractInitial stress (pre-stress) in functionally graded material (FGM) structures is often inevitable because of the limitation of available manufacturing technology. On the basis of the “mechanics of incremental deformations”, the circumferential wave characteristics in FGM cylindrical curved plates under uniform initial stresses in the radial and axial directions are investigated. The Legendre polynomial series method is used to solve the coupled wave equations with variable coefficients. Through numerical examples, the convergence of the polynomial method is discussed. The influences of the initial stresses on the circumferential Lamb-like and the circumferential SH waves are investigated, respectively. Numerical results show that they are quite distinct. Moreover, the influences of the initial stress in the axial direction are very different from those in the radial direction, both on the dispersion curves and on the displacement and stress distributions.

scholarly journals Wave Dispersion in Multilayered Reinforced Nonlocal Plates under Nonlinearly Varying Initial Stress

Eng ◽  
2020 ◽  
Vol 1 (1) ◽  
pp. 31-47
Author(s):  
Mohammad Reza Farajpour ◽  
Ali Reza Shahidi ◽  
Ali Farajpour
Keyword(s):  
Wave Propagation ◽  
Initial Stress ◽  
Volume Fraction ◽  
Wave Dispersion ◽  
Initial Stresses ◽  
Small Scale ◽  
Dependent Model ◽  
Phase And Group Velocities ◽  
The One ◽  
Group Velocities

This paper deals with the effects of initial stress on wave propagations in small-scale plates with shape memory alloy (SMA) nanoscale wires. The initial stress is exerted on the small-scale plate along both in-plane directions. A scale-dependent model of plates is developed for taking into consideration size influences on the wave propagation. In addition, in order to take into account the effects of SMA nanoscale wires, the one-dimensional Brinson’s model is applied. A set of coupled differential equations is obtained for the non-uniformly prestressed small-scale plate with SMA nanoscale wires. An exact solution is obtained for the phase and group velocities of the prestressed small-scale system. The influences of non-uniformly distributed initial stresses as well as scale and SMA effects on the phase and group velocities are explored and discussed. It is found that initial stresses as well as the orientation and volume fraction of SMA nanoscale wires can be used as a controlling factor for the wave propagation characteristics of small-scale plates.

scholarly journals On Rayleigh waves in a thinly layered laminated thermoelastic medium with stress couples under initial stresses

1988 ◽  
Vol 1 (3) ◽  
pp. 161-176
Author(s):  
Pijush Pal Roy ◽  
Lokenath Debnath
Keyword(s):  
Phase Velocity ◽  
Initial Stress ◽  
Rayleigh Waves ◽  
Couple Stress ◽  
Frequency Equation ◽  
Initial Stresses ◽  
Displacement Fields ◽  
Thermoelastic Medium ◽  
Couple Stresses

A study is made of the propagation of Rayleigh waves in a thinly layered laminated thermoelastic medium under deviatoric, hydrostatic, and couple stresses. The frequency equation of the Rayleigh waves is obtained. The phase velocity of the Rayleigh waves depends on the initial stress, deviatoric stress, and the couple stress. The laminated medium is first replaced by an equivalent anisotropic thermoelastic continuum. The corresponding thermoelastic coefficients (after deformation) are derived in terms of initially isotropic thermoelastic coefficients (before deformation) of individual layers. Several particular cases are discussed for the determination of the displacement fields with or without the effect of the couple stress.

scholarly journals Initial stress symmetry and its applications in elasticity

2015 ◽  
Vol 471 (2183) ◽  
pp. 20150448 ◽  
Author(s):  
A. L. Gower ◽  
P. Ciarletta ◽  
M. Destrade
Keyword(s):  
Residual Stress ◽  
Free Energy ◽  
Thermal Expansion ◽  
Initial Stress ◽  
Elastic Deformation ◽  
Deformation Gradient ◽  
Free Energy Density ◽  
Cauchy Stress ◽  
Optimal Target ◽  
External Loads

An initial stress within a solid can arise to support external loads or from processes such as thermal expansion in inert matter or growth and remodelling in living materials. For this reason, it is useful to develop a mechanical framework of initially stressed solids irrespective of how this stress formed. An ideal way to do this is to write the free energy density Ψ in terms of initial stress τ and the elastic deformation gradient F , so we write Ψ = Ψ ( F , τ ). In this paper, we present a new constitutive condition for initially stressed materials, which we call the initial stress symmetry (ISS). We focus on two consequences of this condition. First, we examine how ISS restricts the possible choices of free energy densities Ψ = Ψ ( F , τ ) and present two examples of Ψ that satisfy the ISS. Second, we show that the initial stress can be derived from the Cauchy stress and the elastic deformation gradient. To illustrate, we take an example from biomechanics and calculate the optimal Cauchy stress within an artery subjected to internal pressure. We then use ISS to derive the optimal target residual stress for the material to achieve after remodelling, which links nicely with the notion of homeostasis.

Phantom-Element Technique for Periodic Deformation Analysis of Plain Fabrics Using LS-DYNA

2018 ◽  
Author(s):  
Hikaru Miyaki ◽  
Atsushi Sakuma
Keyword(s):  
Initial Stress ◽  
Periodic Structures ◽  
Woven Fabrics ◽  
Unit Cell ◽  
Woven Fabric ◽  
Initial Stresses ◽  
Deformation Analysis ◽  
Initial State ◽  
Periodic Deformation ◽  
Periodic Analysis

Unit-cell modelling is one of the useful methods to analyze deformation in periodic structures like honeycombs, perforated boards, and woven fabrics. The initial state of the structures is considered to be stress-free in ordinal deformation analysis, but in actual practice, the analysis is difficult, because initial stresses like assembly stress and residual stress need to be considered, as they are known to affect the results. In this study, a technique of taking into account the initial-stress state in woven fabrics is discussed, which has resulted in the establishment of a precise design method for textiles. LS-DYNA, which is a general-purpose finite element (FE) software, has been utilized to simulate the complex deformations in woven fabrics. In this software, a function of the global constraint on boundary conditions facilitates the analysis of periodic structures, but causes difficulties in computing the initial stress states in woven fabrics, as the conditions of mechanical equilibrium have to be satisfied in the governing equations. In particular, duplicated definitions of forced displacement and periodic deformation make the computation impossible, hence, a phantom-element has been introduced to ease the FE analysis by defining these quantities. A unit-cell of the woven fabric is identified and the initial states in stressed conditions can be estimated for periodic structures of plain-woven fabrics by a periodic-analysis technique of LS-DYNA coupled with the phantom-element, which yields a weaving motion of the yarn in plain-woven fabrics.

The Visco-Elastic Behaviors of PVC Coated Fabrics under Different Stress and Temperatures

2010 ◽  
Vol 168-170 ◽  
pp. 1476-1479 ◽  
Author(s):  
Ying Ying Zhang ◽  
Qi Lin Zhang ◽  
Chuan Zhi Zhou
Keyword(s):  
Stress Relaxation ◽  
Initial Stress ◽  
Creep Strain ◽  
Elastic Strain ◽  
Membrane Structure ◽  
Initial Stresses ◽  
Under Five ◽  
Creep And Stress Relaxation ◽  
Coated Fabrics

The visco-elastic behaviors of coated fabrics are important for the design of membrane structure. In this paper, Ferrari 1002 is taken as the research object. The tests of creep and stress relaxation are carried out under five temperatures (23, 40, 50, 60, and 70 ) and three initial stresses (4, 15, and 26 kN/m), respectively. Results show that temperature and initial stress has great effects on the visco-elasticity of coated fabrics. With temperature and initial stress increasing, the visco-elastic behaviors are more obvious. The creep strain can be ignored compared with the corresponding elastic strain. After 24h, the remaining stress is less than 80% the initial stress. This paper can be references for the design of membrane structure.

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