A methodology for obtaining implicit constitutive representations involving the Cauchy stress and the Hencky strain for isotropic materials undergoing a non-dissipative process is developed. Using this methodology, a general constitutive representation for a subclass of implicit models relating the Cauchy stress and the Hencky strain is obtained for an isotropic material with no internal constraints. It is shown that even for this subclass, unlike classical Green elasticity, one has to specify three potentials to relate the Cauchy stress and the Hencky strain. Then, a procedure to obtain implicit constitutive representations for isotropic materials with internal constraints is presented. As an illustration, it is shown that for incompressible materials the Cauchy stress and the Hencky strain could be related through a single potential. Finally, constitutive approximations are obtained when the displacement gradient is small. © 2017, Springer International Publishing AG.

Umakanthan Saravanan

Department Of Civil Engineering

Mathematical techniques

COMPRESSIBLE

HENCKY STRAINS

INCOMPRESSIBLE

Isotropic

NON-DISSIPATIVE

Incompressible flow

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IIT Madras

Zeitschrift fur Angewandte Mathematik und Physik

This work documents the formulation of an axial element that could handle implicit constitutive relations. This formulation uses nodal displacements, member forces, and reaction forces as primary unknowns. For some assumed values of these primary unknowns satisfying the nodal equilibrium equations, the computed stretch and stress in the members need not be in agreement with the constitutive relation. Hence, these primary unknowns are found such that the error in the constitutive relation for the members are minimized subjected to the constraint that the nodal equilibrium equations hold. Axial element developed using this concept is used to analyze a few trusses. The results of this formulation are in agreement with the results obtained from ABAQUS ® when stress is an explicit function of the stretch ratio. Also, some statically determinate as well as indeterminate trusses are solved for an implicit constitutive relation and the reasonableness of the obtained results established. © 2019 Elsevier Ltd

International Journal of Non-Linear Mechanics

Large deformation axial element for implicit constitutive relations

The validity of some of the assumptions made during systematic development of constitutive relation – homogeneity, isotropy, and non-dissipative response – is examined experimentally by testing plain concrete cylinders in the circumferential displacement controlled uniaxial compression tests. The tested cylinders indicate that the surface strains are not uniform and that the principal direction of the strain varies with the magnitude of the applied load. Hence, these cylinders are either deforming from a stressed state or anisotropic or both. The percentage dissipation computed as the area between the loading and unloading curves normalized using the area under the unloading curve is less than 10 percent up to 90 percent of the peak load. Therefore the mechanical response of concrete can be considered as non-dissipative. Assuming that the magnitude of the residual stresses present in the concrete panels would be small compared to the stresses arising due to the applied load, these residual stresses are ignored. Consequently, using the implicit constitutive theory framework for isotropic and compressible materials undergoing a non-dissipative process from a stress-free reference configuration, a three-dimensional constitutive relation for plain concrete is proposed. The material parameters in the proposed model are estimated from the uniaxial compression test and equal biaxial compression test. These parameters are expressed using initial Young's modulus, initial Poisson's ratio, maximum uniaxial compressive stress, the axial and transverse strain corresponding to the maximum uniaxial compressive stress, maximum equal biaxial compressive stress and the equal biaxial strain corresponding to maximum equal biaxial compressive stress. The ability of the proposed model to capture the response of various grade and type of concrete in ten biaxial experiments reported in the literature is examined. It is found that despite the limiting assumptions, the compression-compression, compression-tension and tension-tension stress strain response of the concrete in these ten experiments are adequately captured by the proposed model with mean R 2 value of 0.94. © 2019 Elsevier Ltd

Construction and Building Materials

A three dimensional constitutive model for plain cement concrete

Using a custom built experimental setup, biaxial and uniaxial extension tests are performed on the same specimen. In biaxial tests, the stretch ratio along y direction is held constant and the stretch ratio in the perpendicular x direction is increased gradually from 1 to 1.4. Stored energy function is obtained from seven different biaxial tests performed by varying the stretch ratio along the y direction at which it is held fixed. Stored energy functions are proposed for the tested rubber sheets by allowing the stored energy to be a function of the principal invariants of right Cauchy-Green deformation tensor or the invariants of the Hencky strain or the principal stretch ratios. To evaluate the performance of the developed stored energy functions, they are compared with 10 existing constitutive relations available in literature in terms of their ability to capture their biaxial response and predicting capability for the uniaxial experiments on the rubber specimens studied. This study reveals that the stored energy function developed in this study as a function of the principal invariants of the right Cauchy-Green deformation tensor predicts the uniaxial behavior the best and capture the biaxial response second best among the models studied for the tested material. However, even this model on an average has 25 percent error in the predicted stresses. © 2015 Elsevier Ltd. All rights reserved.

Mechanics of Materials

A new set of biaxial and uniaxial experiments on vulcanized rubber and attempts at modeling it using classical hyperelastic models

Zeitschrift für angewandte Mathematik und Physik

Large deformations of a new class of incompressible elastic bodies

On stress-based piecewise elasticity for limited strain extensibility materials

Continuum Mechanics and Thermodynamics

On the consequences of the constraint of incompressibility with regard to a new class of constitutive relations for elastic bodies: small displacement gradient approximation

Acta Mechanica

New compressible hyper-elastic models for rubberlike materials

Representations for implicit constitutive relations describing non-dissipative response of isotropic materials

Journal	Zeitschrift fur Angewandte Mathematik und Physik
Publisher	Birkhauser Verlag AG
ISSN	00442275
Open Access	No