scholarly journals Sharp conditions for the linearization of finite elasticity

2021 ◽  
Vol 60 (5) ◽  
Author(s):  
Edoardo Mainini ◽  
Danilo Percivale
Keyword(s):  
Elastic Energy ◽  
Compatibility Condition ◽  
Finite Elasticity ◽  
Linear Elastic ◽  
Approximating Sequence ◽  
Standard Linear ◽  
Nonlinear Elastic ◽  
External Loads ◽  
Variational Limit

AbstractWe consider the topic of linearization of finite elasticity for pure traction problems. We characterize the variational limit for the approximating sequence of rescaled nonlinear elastic energies. We show that the limiting minimal value can be strictly lower than the minimal value of the standard linear elastic energy if a strict compatibility condition for external loads does not hold. The results are provided for both the compressible and the incompressible case.

scholarly journals Variational linearization of pure traction problems in incompressible elasticity

2020 ◽  
Vol 71 (5) ◽  
Author(s):  
Edoardo Mainini ◽  
Danilo Percivale
Keyword(s):  
Finite Elasticity ◽  
Linearized Elasticity ◽  
Incompressible Elasticity ◽  
External Loads ◽  
Variational Limit

Abstract We consider pure traction problems, and we show that incompressible linearized elasticity can be obtained as variational limit of incompressible finite elasticity under suitable conditions on external loads.

scholarly journals A nonlinear elastic description of cell preferential orientations over a stretched substrate

2021 ◽  
Author(s):  
Giulio Lucci ◽  
Luigi Preziosi
Keyword(s):  
Linear Relationship ◽  
Finite Elasticity ◽  
Strain Tensor ◽  
Orientation Angle ◽  
Nonlinear Effects ◽  
Major Axis ◽  
Elastic Model ◽  
Linear Elastic ◽  
Experimental Findings ◽  
Nonlinear Elastic

AbstractThe active response of cells to mechanical cues due to their interaction with the environment has been of increasing interest, since it is involved in many physiological phenomena, pathologies, and in tissue engineering. In particular, several experiments have shown that, if a substrate with overlying cells is cyclically stretched, they will reorient to reach a well-defined angle between their major axis and the main stretching direction. Recent experimental findings, also supported by a linear elastic model, indicated that the minimization of an elastic energy might drive this reorientation process. Motivated by the fact that a similar behaviour is observed even for high strains, in this paper we address the problem in the framework of finite elasticity, in order to study the presence of nonlinear effects. We find that, for a very large class of constitutive orthotropic models and with very general assumptions, there is a single linear relationship between a parameter describing the biaxial deformation and $$\cos ^2\theta _{\mathrm{eq}}$$ cos 2 θ eq , where $$\theta _{\mathrm{eq}}$$ θ eq is the orientation angle of the cell, with the slope of the line depending on a specific combination of four parameters that characterize the nonlinear constitutive equation. We also study the effect of introducing a further dependence of the energy on the anisotropic invariants related to the square of the Cauchy–Green strain tensor. This leads to departures from the linear relationship mentioned above, that are again critically compared with experimental data.

An Appraisal of the Paper by O'Carrol et al. (1990)

1995 ◽  
Vol 209 (3) ◽  
pp. 203-205 ◽  
Author(s):  
A Strozzi ◽  
A Unsworth
Keyword(s):  
Finite Element ◽  
Finite Elasticity ◽  
Experimental Measurements ◽  
Linear Elastic ◽  
Numerical Predictions ◽  
Elasticity Effects

The paper by O'Carrol et al. (1), which addresses the problem of an elastomeric disc indented by a spherical punch, has been evaluated. The sources of disagreement between linear elastic numerical predictions and experimental measurements noted in this paper have been critically examined in the light of finite element forecasts obtained with a package which incorporates finite elasticity effects and incompressibility.

scholarly journals Relaxation of nonlinear elastic energies involving the deformed configuration and applications to nematic elastomers

2019 ◽  
Vol 25 ◽  
pp. 19 ◽  
Author(s):  
Carlos Mora-Corral ◽  
Marcos Oliva
Keyword(s):  
Sobolev Space ◽  
Elastic Energy ◽  
Lower Semicontinuous ◽  
Deformation Gradient ◽  
Variational Model ◽  
Nematic Elastomer ◽  
Nematic Elastomers ◽  
Lower Semicontinuous Envelope ◽  
Nonlinear Elastic

We start from a variational model for nematic elastomers that involves two energies: mechanical and nematic. The first one consists of a nonlinear elastic energy which is influenced by the orientation of the molecules of the nematic elastomer. The nematic energy is an Oseen–Frank energy in the deformed configuration. The constraint of the positivity of the determinant of the deformation gradient is imposed. The functionals are not assumed to have the usual polyconvexity or quasiconvexity assumptions to be lower semicontinuous. We instead compute its relaxation, that is, the lower semicontinuous envelope, which turns out to be the quasiconvexification of the mechanical term plus the tangential quasiconvexification of the nematic term. The main assumptions are that the quasiconvexification of the mechanical term is polyconvex and that the deformation is in the Sobolev space W1,p (with p > n − 1 and n the dimension of the space) and does not present cavitation.

scholarly journals Nonlinear Poisson effects in soft honeycombs

2014 ◽  
Vol 470 (2169) ◽  
pp. 20140363 ◽  
Author(s):  
L. Angela Mihai ◽  
Alain Goriely
Keyword(s):  
Poisson Ratio ◽  
Finite Elasticity ◽  
Tensile Load ◽  
Honeycomb Structure ◽  
Wall Material ◽  
Uniaxial Tensile ◽  
Cell Level ◽  
Standard Geometry ◽  
Local Cell ◽  
Nonlinear Elastic

We examine solid cellular structures within the theoretical framework of finite elasticity, whereby we assume that the cell wall material is nonlinear elastic. This enables us to identify new mechanical effects that appear in cellular materials when elastically deformed, and to explore the physical properties that influence them. We find that, when a honeycomb structure of hyperelastic material and standard geometry, such as rectangular-, hexagonal- or diamond-shaped cells, contains walls which are inclined relative to an applied uniaxial tensile load, these walls tend to expand both in the direction of the load and in the perpendicular direction, producing an apparent negative Poisson's ratio at local cell level. Moreover, we show that this (negative) Poisson ratio decreases as the magnitude of the tensile load increases. For these structures, Poisson's ratios greater than 0.5 are obtained in uniaxial compression. Similar effects in structures with linearly elastic cell walls do not occur.

On the Applicability of Sneddon's Solution for Interpreting the Indentation of Nonlinear Elastic Biopolymers

2014 ◽  
Vol 81 (9) ◽  
Author(s):  
Man-Gong Zhang ◽  
Jinju Chen ◽  
Xi-Qiao Feng ◽  
Yanping Cao
Keyword(s):  
Elastic Properties ◽  
Viscoelastic Properties ◽  
Contact Theory ◽  
Elastic Contact ◽  
Flat Punch ◽  
Linear Elastic ◽  
Contact Models ◽  
Properties Of Materials ◽  
Nonlinear Elastic ◽  
Viscoelastic Contact

Indentation has been widely used to characterize the mechanical properties of biopolymers. Besides Hertzian solution, Sneddon's solution is frequently adopted to interpret the indentation data to deduce the elastic properties of biopolymers, e.g., elastic modulus. Sneddon's solution also forms the basis to develop viscoelastic contact models for determining the viscoelastic properties of materials from either conical or flat punch indentation responses. It is worth mentioning that the Sneddon's solution was originally proposed on the basis of linear elastic contact theory. However, in both conical and flat punch indentation of compliant materials, the indented solid may undergo finite deformation. In this case, the extent to which the Sneddon's solution is applicable so far has not been systematically investigated. In this paper, we use the combined theoretical, computational, and experimental efforts to investigate the indentation of hyperelastic compliant materials with a flat punch or a conical tip. The applicability of Sneddon's solutions is examined. Furthermore, we present new models to determine the elastic properties of nonlinear elastic biopolymers.

scholarly journals Random Fiber Networks With Superior Properties Through Network Topology Control

2019 ◽  
Vol 86 (8) ◽  
Author(s):  
S. Deogekar ◽  
Z. Yan ◽  
R. C. Picu
Keyword(s):  
Network Architecture ◽  
Elastic Behavior ◽  
Fiber Networks ◽  
Linear Elastic ◽  
New Class ◽  
Link Density ◽  
Cross Link Density ◽  
Random Fiber Networks ◽  
Nonlinear Elastic ◽  
Poisson Contraction

In this work, we study the effect of network architecture on the nonlinear elastic behavior and strength of athermal random fiber networks of cellular type. We introduce a topology modification of Poisson–Voronoi (PV) networks with convex cells, leading to networks with stochastic nonconvex cells. Geometric measures are developed to characterize this new class of nonconvex Voronoi (NCV) networks. These are softer than the reference PV networks at the same nominal network parameters such as density, cross-link density, fiber diameter, and connectivity number. Their response is linear elastic over a broad range of strains, unlike PV networks that exhibit a gradual increase of the tangent stiffness starting from small strains. NCV networks exhibit much smaller Poisson contraction than any network of same nominal parameters. Interestingly, the strength of NCV networks increases continuously with an increasing degree of nonconvexity of the cells. These exceptional properties render this class of networks of interest in a variety of applications, such as tissue scaffolds, nonwovens, and protective clothing.

Analysis of Indentation: Implications for Measuring Mechanical Properties With Atomic Force Microscopy

1999 ◽  
Vol 121 (5) ◽  
pp. 462-471 ◽  
Author(s):  
K. D. Costa ◽  
F. C. P. Yin
Keyword(s):  
Indentation Depth ◽  
Constitutive Law ◽  
Elastic Materials ◽  
Force Microscopy ◽  
Linear Elastic ◽  
Afm Indentation ◽  
Atomic Force ◽  
Afm Probe ◽  
Nonlinear Elastic ◽  
Indenter Geometry

Indentation using the atomic force microscope (AFM) has potential to measure detailed micromechanical properties of soft biological samples. However, interpretation of the results is complicated by the tapered shape of the AFM probe tip, and its small size relative to the depth of indentation. Finite element models (FEMs) were used to examine effects of indentation depth, tip geometry, and material nonlinearity and heterogeneity on the finite indentation response. Widely applied infinitesimal strain models agreed with FEM results for linear elastic materials, but yielded substantial errors in the estimated properties for nonlinear elastic materials. By accounting for the indenter geometry to compute an apparent elastic modulus as a function of indentation depth, nonlinearity and heterogeneity of material properties may be identified. Furthermore, combined finite indentation and biaxial stretch may reveal the specific functional form of the constitutive law—a requirement for quantitative estimates of material constants to be extracted from AFM indentation data.

Evaluation of Stress-Controlled Permeability of Coal Fractures - A Laboratory Study

2013 ◽  
Vol 734-737 ◽  
pp. 703-708
Author(s):  
Yi Dong Cai ◽  
Da Meng Liu ◽  
Yan Bin Yao ◽  
Bai Ren Zhang ◽  
Jun Qian Li ◽  
...  
Keyword(s):  
Elastic Deformation ◽  
Axial Stress ◽  
Fractal Dimensions ◽  
Confining Pressure ◽  
Control Factor ◽  
Type I ◽  
Permeability Characteristics ◽  
Linear Elastic ◽  
Variable Permeability ◽  
Nonlinear Elastic

Experiments on coal permeability with saturated water under tri-axial stress were conducted. The relationship between stress and permeability under tri-axial stress was analyzed on the rock mechanical experimental rig (GAW-2000). After the experiments on permeability, the fracture characteristics were researched by X-ray computerized tomography, which shows that the bituminous coal normally has high fractal dimensions (generally over 1.8) and wide aperture. The results for permeability reveal that bituminous coals always have variable permeability characteristics under incremental axial stress due to its inherent fracture features. It can be divided into two types: type I, at the linear and nonlinear elastic deformation and peak stage, the permeability keeps rising, which is represented by FYGY8 #. The main control factor of permeability should be related to coal microfractures and coal compositions. Type II, which is represented by sample YCLZ2#, in the initial linear elastic stage, there is a decrease trend in the permeability performance, and then permeability gradually rise when it comes into the stage of nonlinear elastic deformation. The permeability will keep go down after coal becomes soften under the action of confining pressure, compaction.

scholarly journals Large Displacement J-Integral Double Cantilever Beam (DCB) Test Method for Mode I Fracture Toughness

2020 ◽  
Author(s):  
Joshua Gunderson
Keyword(s):  
Fracture Toughness ◽  
Double Cantilever Beam ◽  
Test Method ◽  
Large Displacement ◽  
Mode I ◽  
Mode I Fracture ◽  
Mode I Fracture Toughness ◽  
Linear Elastic ◽  
True Value ◽  
Nonlinear Elastic

The J-integral is used to develop an alternative double cantilever beam (DCB) test method for the Mode I fracture toughness suitable for both small and large displacements. The current focus is the experimental determination of the Mode I interlaminar fracture toughness of composite materials, but the method is generally applicable to other similar tests and material systems, such as to the Mode I fracture toughness of adhesives. A series of five identical specimens are tested to compare the linear-elastic fracture mechanics method recommended by ASTM, which makes use of linear beam theory with root rotation, large displacement, and end block corrections, with the new nonlinear-elastic and elastic-plastic fracture mechanics method, which does not require these corrections. Experimental results show excellent agreement between the two methods over a series of five tests of primarily linear-elastic DCB specimens subjected to moderate to large displacements as defined in the ASTM standard. Furthermore, an agreement is found between the results of the derivations for the two methods being compared, whereby the large displacement equation for JIc presented in this work is identical to the equation given by J. G. Williams (1987) and which he found to be the true value of GIc. It is the true value of GIc that the large displacement and root rotation correction factors were intended to approximate, and the test method presented here allows for direct measurement of its parameters and evaluation. This method has the added benefit that specimens can be primarily linear-elastic or nonlinear-elastic at the crack tip and may extend to those that are elastic-plastic at the crack tip.

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