scholarly journals Ordered Rate Constitutive Theories for Non-Classical Thermofluids Based on Convected Time Derivatives of the Strain and Higher Order Rotation Rate Tensors Using Entropy Inequality

Entropy ◽  
2020 ◽  
Vol 22 (4) ◽  
pp. 443
Author(s):  
Karan S. Surana ◽  
Stephen W. Long
Keyword(s):  
Representation Theorem ◽  
Moment Tensor ◽  
Strain Tensor ◽  
Entropy Inequality ◽  
Cauchy Stress ◽  
Symmetric Part ◽  
Gradient Tensor ◽  
Constitutive Theories ◽  
Derivatives Of ◽  
Ordered Rate Constitutive Theories

This paper considers non-classical continuum theory for thermoviscous fluids without memory incorporating internal rotation rates resulting from the antisymmetric part of the velocity gradient tensor to derive ordered rate constitutive theories for the Cauchy stress and the Cauchy moment tensor based on entropy inequality and representation theorem. Using the generalization of the conjugate pairs in the entropy inequality, the ordered rate constitutive theory for Cauchy stress tensor considers convected time derivatives of the Green’s strain tensor (or Almansi strain tensor) of up to orders n ε as its argument tensors and the ordered rate constitutive theory for the Cauchy moment tensor considers convected time derivatives of the symmetric part of the rotation gradient tensor up to orders n Θ . While the convected time derivatives of the strain tensors are well known the convected time derivatives of higher orders of the symmetric part of the rotation gradient tensor need to be derived and are presented in this paper. Complete and general constitutive theories based on integrity using conjugate pairs in the entropy inequality and the generalization of the argument tensors of the constitutive variables and the representation theorem are derived and the material coefficients are established. It is shown that for the type of non-classical thermofluids considered in this paper the dissipation mechanism is an ordered rate mechanism due to convected time derivatives of the strain tensor as well as the convected time derivatives of the symmetric part of the rotation gradient tensor. The derivations of the constitutive theories presented in the paper is basis independent but can be made basis specific depending upon the choice of the specific basis for the constitutive variables and the argument tensors. Simplified linear theories are also presented as subset of the general constitutive theories and are compared with published works.

Ordered Rate Constitutive Theories for Non-Classical Thermoviscoelastic Fluids Incorporating Internal and Cosserat Rotation Rates

2018 ◽  
Vol 10 (02) ◽  
pp. 1850012 ◽  
Author(s):  
K. S. Surana ◽  
A. D. Joy ◽  
J. N. Reddy
Keyword(s):  
Internal Rotation ◽  
Velocity Gradient ◽  
Cauchy Stress ◽  
Balance Laws ◽  
Gradient Tensor ◽  
Moment Tensors ◽  
Velocity Gradient Tensor ◽  
Constitutive Theories ◽  
Rotation Rates ◽  
Derivatives Of

This paper considers conservation and balance laws for non-classical fluent continua in the presence of internal rotation rates due to the velocity gradient tensor and the rotation rates due to Cosserat rotations. In these balance laws, the internal rotation rates are completely defined as functions of the velocity gradient tensor, but the Cosserat rotation rates are additional three degrees of freedom at each material point. When these rotation rates are resisted by the deforming continua, conjugate moments are created. For thermoviscoelastic fluent continua, these result in additional dissipation mechanism as well as rheology. This paper presents a thermodynamically consistent derivation of constitutive theories for such fluids based on the entropy inequality in conjunction with the representation theorem using integrity, i.e., complete basis. Material coefficients are derived and discussed. The constitutive theories are presented in the absence as well as presence of the balance of moments of moments as a balance law and are compared with the classical continuum theories. Retardation moduli corresponding to the Cauchy stress tensor, both symmetric and antisymmetric, as well as the symmetric and antisymmetric Cauchy moment tensors are derived. The constitutive theories presented in this paper are ordered rate theories. These incorporate convected time derivatives of the strain tensor up to order [Formula: see text] and the convected time derivatives of the stress and moment tensors up to certain orders. Simplified forms of the constitutive theories are also derived. It is shown and concluded that the constitutive theories for thermoviscoelastic fluids based on classical continuum theory such as Maxwell model, Oldroyd-B model, Giesekus model, etc., are all a subset of the constitutive theories presented in this paper for non-classical physics. Both compressible and incompressible thermoviscoelastic fluids are considered.

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.

scholarly journals Work conjugacy error in commercial finite-element codes: its magnitude and how to compensate for it

2012 ◽  
Vol 468 (2146) ◽  
pp. 3047-3058 ◽  
Author(s):  
Zdeněk P. Bažant ◽  
Mahendra Gattu ◽  
Jan Vorel
Keyword(s):  
Finite Element ◽  
Finite Strain ◽  
Strain Tensor ◽  
Biological Tissues ◽  
Organic Soils ◽  
Osteoporotic Bone ◽  
Cauchy Stress ◽  
Ceramic Foams ◽  
Commercial Program ◽  
Updated Lagrangian

Most commercial finite-element programs use the Jaumann (or co-rotational) rate of Cauchy stress in their incremental (Riks) updated Lagrangian loading procedure. This rate was shown long ago not to be work-conjugate with the Hencky (logarithmic) finite strain tensor used in these programs, nor with any other finite strain tensor. The lack of work-conjugacy has been either overlooked or believed to cause only negligible errors. Presented are examples of indentation of a naval-type sandwich plate with a polymeric foam core, in which the error can reach 28.8 per cent in the load and 15.3 per cent in the work of load (relative to uncorrected results). Generally, similar errors must be expected for all highly compressible materials, such as metallic and ceramic foams, honeycomb, loess, silt, organic soils, pumice, tuff, osteoporotic bone, light wood, carton and various biological tissues. It is shown that a previously derived equation relating the tangential moduli tensors associated with the Jaumann rates of Cauchy and Kirchhoff stresses can be used in the user’s material subroutine of a black-box commercial program to cancel the error due to the lack of work-conjugacy and make the program perform exactly as if the Jaumann rate of Kirchhoff stress, which is work-conjugate, were used.

Moire´ Patterns of Partial Derivatives of Displacement Components

1966 ◽  
Vol 33 (4) ◽  
pp. 901-906 ◽  
Author(s):  
V. J. Parks ◽  
A. J. Durelli
Keyword(s):  
Nonlinear Deformation ◽  
Strain Tensor ◽  
Strain Analysis ◽  
The Other ◽  
Partial Derivatives ◽  
Moire Patterns ◽  
Derivatives Of ◽  
Moiré Patterns

The knowledge of the partial derivatives of displacement components is essential in strain analysis. Two methods of determining these partial derivatives using moire´ effects are presented in this paper. One consists of superposing two shifted copies of the same deformed grating of lines. The other consists of superposing two shifted copies of moire´ patterns of displacement components. Explanations of the phenomena, based on the parametric properties of superposed families of lines, are given. Applications to the determination of the nonlinear deformation and strain tensor are included.

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