scholarly journals Verification of Continuum Mechanics Predictions with Experimental Mechanics

Materials ◽  
2019 ◽  
Vol 13 (1) ◽  
pp. 77
Author(s):  
Cesar A. Sciammarella ◽  
Luciano Lamberti ◽  
Federico M. Sciammarella
Keyword(s):  
Stress Tensor ◽  
Continuum Mechanics ◽  
Strain Tensor ◽  
Scalar Fields ◽  
Identification Problem ◽  
Theoretical Concept ◽  
Cauchy Stress ◽  
Cauchy Stress Tensor ◽  
Mathematical Functions ◽  
Eulerian Derivative

The general goal of the study is to connect theoretical predictions of continuum mechanics with actual experimental observations that support these predictions. The representative volume element (RVE) bridges the theoretical concept of continuum with the actual discontinuous structure of matter. This paper presents an experimental verification of the RVE concept. Foundations of continuum kinematics as well as mathematical functions relating displacement vectorial fields to the recording of these fields by a light sensor in the form of gray-level scalar fields are reviewed. The Eulerian derivative field tensors are related to the deformation of the continuum: the Euler–Almansi tensor is extracted, and its properties are discussed. The compatibility between the Euler–Almansi tensor and the Cauchy stress tensor is analyzed. In order to verify the concept of the RVE, a multiscale analysis of an Al–SiC composite material is carried out. Furthermore, it is proven that the Euler–Almansi strain tensor and the Cauchy stress tensor are conjugate in the Hill–Mandel sense by solving an identification problem of the constitutive model of urethane rubber.

ON THE SYSTEMATICS OF ENERGETIC TERMS IN CONTINUUM MECHANICS, AND A NOTE ON GIBBS (1877)

2008 ◽  
Vol 22 (27) ◽  
pp. 4863-4876 ◽  
Author(s):  
FALK H. KOENEMANN
Keyword(s):  
Energy Conservation ◽  
Stress Tensor ◽  
Elastic Deformation ◽  
Continuum Mechanics ◽  
Conservation Law ◽  
Cauchy Stress ◽  
Cauchy Stress Tensor ◽  
First Law Of Thermodynamics ◽  
Energetic Properties ◽  
Energy Conservation Law

The systematics of energetic terms as they are taught in continuum mechanics deviate seriously from the standard doctrine in physics, resulting in a profound misconception. It is demonstrated that the First Law of Thermodynamics has been routinely reinterpreted in a sense that would make it subordinate to Bernoulli's energy conservation law. Proof is given to the effect that the Cauchy stress tensor does not exist. Furthermore, it is shown that the attempt by Gibbs to find a thermodynamic understanding for elastic deformation does not sufficiently account for all the energetic properties of such a process.

scholarly journals A Generalized Method for the Analysis of Planar Biaxial Mechanical Data Using Tethered Testing Configurations

2015 ◽  
Vol 137 (6) ◽  
Author(s):  
Will Zhang ◽  
Yuan Feng ◽  
Chung-Hao Lee ◽  
Kristen L. Billiar ◽  
Michael S. Sacks
Keyword(s):  
Stress Tensor ◽  
Test Data ◽  
Soft Tissues ◽  
Mechanical Test ◽  
Specimen Geometry ◽  
Parameter Determination ◽  
Biaxial Test ◽  
Cauchy Stress ◽  
Cauchy Stress Tensor ◽  
Thin Sections

Simulation of the mechanical behavior of soft tissues is critical for many physiological and medical device applications. Accurate mechanical test data is crucial for both obtaining the form and robust parameter determination of the constitutive model. For incompressible soft tissues that are either membranes or thin sections, planar biaxial mechanical testing configurations can provide much information about the anisotropic stress–strain behavior. However, the analysis of soft biological tissue planar biaxial mechanical test data can be complicated by in-plane shear, tissue heterogeneities, and inelastic changes in specimen geometry that commonly occur during testing. These inelastic effects, without appropriate corrections, alter the stress-traction mapping and violates equilibrium so that the stress tensor is incorrectly determined. To overcome these problems, we presented an analytical method to determine the Cauchy stress tensor from the experimentally derived tractions for tethered testing configurations. We accounted for the measured testing geometry and compensate for run-time inelastic effects by enforcing equilibrium using small rigid body rotations. To evaluate the effectiveness of our method, we simulated complete planar biaxial test configurations that incorporated actual device mechanisms, specimen geometry, and heterogeneous tissue fibrous structure using a finite element (FE) model. We determined that our method corrected the errors in the equilibrium of momentum and correctly estimated the Cauchy stress tensor. We also noted that since stress is applied primarily over a subregion bounded by the tethers, an adjustment to the effective specimen dimensions is required to correct the magnitude of the stresses. Simulations of various tether placements demonstrated that typical tether placements used in the current experimental setups will produce accurate stress tensor estimates. Overall, our method provides an improved and relatively straightforward method of calculating the resulting stresses for planar biaxial experiments for tethered configurations, which is especially useful for specimens that undergo large shear and exhibit substantial inelastic effects.

Fractional Derivative Constitutive Models for Finite Deformation of Viscoelastic Materials

2015 ◽  
Vol 10 (6) ◽  
Author(s):  
Masataka Fukunaga ◽  
Nobuyuki Shimizu
Keyword(s):  
Fractional Derivative ◽  
Stress Tensor ◽  
Finite Deformation ◽  
Dynamical Behavior ◽  
Strain Tensor ◽  
Constitutive Models ◽  
Cauchy Stress ◽  
Viscoelastic Materials ◽  
Kirchhoff Stress Tensor ◽  
Biot Stress

A methodology to derive fractional derivative constitutive models for finite deformation of viscoelastic materials is proposed in a continuum mechanics treatment. Fractional derivative models are generalizations of the models given by the objective rates. The method of generalization is applied to the case in which the objective rate of the Cauchy stress is given by the Truesdell rate. Then, a fractional derivative model is obtained in terms of the second Piola–Kirchhoff stress tensor and the right Cauchy-Green strain tensor. Under the assumption that the dynamical behavior of the viscoelastic materials comes from a complex combination of elastic and viscous elements, it is shown that the strain energy of the elastic elements plays a fundamental role in determining the fractional derivative constitutive equation. As another example of the methodology, a fractional constitutive model is derived in terms of the Biot stress tensor. The constitutive models derived in this paper are compared and discussed with already existing models. From the above studies, it has been proved that the methodology proposed in this paper is fully applicable and effective.

Some topics on a new class of elastic bodies

2009 ◽  
Vol 465 (2105) ◽  
pp. 1377-1392 ◽  
Author(s):  
Roger Bustamante
Keyword(s):  
Constitutive Equation ◽  
Stress Tensor ◽  
Cauchy Stress ◽  
Cauchy Stress Tensor ◽  
Weak Formulation ◽  
New Class ◽  
Elastic Bodies ◽  
Small Deformations

In this paper, we study the problem of prescribing deformation as a function of stresses. For the particular case of small deformations, we find a weak formulation, from which we define the constitutive equation of a Green-like material, where an energy function that depends on the Cauchy stress tensor is proposed. Constraints on the deformation are studied for this new class of elastic bodies.

scholarly journals Exact and linearized refractive index stress-dependence in anisotropic photoelastic crystals

2020 ◽  
Vol 476 (2238) ◽  
pp. 20190854
Author(s):  
Fabrizio Daví
Keyword(s):  
Refractive Index ◽  
Stress Tensor ◽  
Permittivity Tensor ◽  
Nonlinear Dependence ◽  
Cauchy Stress ◽  
Square Root ◽  
Cauchy Stress Tensor ◽  
Starting Point ◽  
Higher Order Terms ◽  
Photoelastic Analysis

For the permittivity tensor of photoelastic anisotropic crystals, we obtain the exact nonlinear dependence on the Cauchy stress tensor. We obtain the same result for its square root, whose principal components, the crystal principal refractive index, are the starting point for any photoelastic analysis of transparent crystals. From these exact results we then obtain, in a totally general manner, the linearized expressions to within higher-order terms in the stress tensor for both the permittivity tensor and its square root. We finish by showing some relevant examples of both nonlinear and linearized relations for optically isotropic, uniaxial and biaxial crystals.

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