First Piola-Kirchhoff Stress

2008 ◽  
pp. 1573-1573
Keyword(s):  
Kirchhoff Stress

Nonuniform Stress Field Determination Based on Deformation Measurement

2021 ◽  
pp. 1-26
Author(s):  
Cheng Liu
Keyword(s):  
Principal Stress ◽  
Deformation Gradient ◽  
Deformation Measurement ◽  
Cauchy Stress ◽  
Kirchhoff Stress ◽  
Nonuniform Deformation ◽  
Principal Stretch ◽  
Stress Orientation ◽  
Lagrangian Strain ◽  
Stretch Orientation

Abstract We demonstrate a technique that, under certain circumstances, will determine stresses associated with a nonuniform deformation field without knowing the detailed constitutive behavior of the deforming material. This technique is based on (1) a detailed deformation measurement of a domain and (2) the observation that for isotropic materials, the strain and the stress, which form the so-called work-conjugate pair, are co-axial, or their eigenvectors share the same direction. The particular measures for strain and stress considered are the Lagrangian strain and the second Piola-Kirchhoff stress. The deformation measurement provides the field of the principal stretch orientation θλ and since the Lagrangian strain and the second Piola-Kirchhoff stress are co-axial, the principal stress orientation θs of the second Piola-Kirchhoff stress is determined. The Cauchy stress is related to the second Piola-Kirchhoff stress through the deformation gradient tensor, which can be measured experimentally. We then show that the principal stress orientation θσ of the Cauchy stress is the sum of the principal stretch orientation θλ and the local rigid-body rotation θq, which is determinable by the deformation gradient through polar decomposition. With the principal stress orientation θσ known, the equation of equilibrium, now in terms of the two principal stresses σ1 and σ2, and θσ, can be solved numerically with appropriate traction boundary conditions. The technique is then applied to the experimental case of nonuniform deformation of a PVC sheet with a circular hole and subject to tension. Limitations and restrictions of the technique and possible extensions will be discussed.

scholarly journals Correction to: An efficient and stabilised SPH method for large strain metal plastic deformations

2019 ◽  
Vol 7 (3) ◽  
pp. 541-541
Author(s):  
Giorgio Greto ◽  
Sivakumar Kulasegaram
Keyword(s):  
Stress Tensor ◽  
Large Strain ◽  
Sph Method ◽  
Plastic Deformations ◽  
Kirchhoff Stress ◽  
Time Stepping ◽  
Kirchhoff Stress Tensor

The symbol was introduced incorrectly inside the “Time-stepping the solution” box, directly under the “Compute first Piola–Kirchhoff stress tensor Pi” as in “Appendix A” listing.

scholarly journals Covariance in plasticity

2013 ◽  
Vol 469 (2156) ◽  
pp. 20130073 ◽  
Author(s):  
Paolo Maria Mariano
Keyword(s):  
Second Law Of Thermodynamics ◽  
Second Law ◽  
Kirchhoff Stress ◽  
Perfectly Plastic ◽  
Unique Source ◽  
Elastic Perfectly Plastic

Covariance is imposed to the second law of thermodynamics and consequences are shown for elastic-perfectly plastic bodies. In this setting, pointwise balances of standard and configurational actions, constitutive restrictions on the first Piola–Kirchhoff stress and the Eshelby one, and the structure of the dissipation are all derived from a unique source.

A Rate-Independent Constitutive Theory for Finite Inelastic Deformation

1987 ◽  
Vol 54 (1) ◽  
pp. 15-21 ◽  
Author(s):  
M. M. Carroll
Keyword(s):  
Porous Materials ◽  
Inelastic Deformation ◽  
Constitutive Theory ◽  
Ideal Behavior ◽  
Kirchhoff Stress ◽  
Loading Function ◽  
Microstructural Effects ◽  
Lagrangian Strain

A rate-independent constitutive theory for finite inelastic deformation is formulated in terms of the symmetric Piola-Kirchhoff stress, the Lagrangian strain, and a kinematic tensor which describes inelastic or microstructural effects. Assumptions of (a) continuity in the transition from loading to neutral loading, (b) consistency, and (c) nonnegative work in closed cycles of deformation, lead to simplification of the theory. The response is described by two scalar functions — a stress potential and a loading function. The theory can describe isotropic or anisotropic response, and allows for hardening, softening, or ideal behavior. It may also be appropriate to describe the response of porous materials, such as metals, rocks and ceramics, and also the evolution of damage.

Energy-Conservation Error Due to Use of Green–Naghdi Objective Stress Rate in Commercial Finite-Element Codes and Its Compensation

2013 ◽  
Vol 81 (2) ◽  
Author(s):  
Zdeňek P. Bažant ◽  
Jan Vorel
Keyword(s):  
Finite Element ◽  
Energy Conservation ◽  
Strain Tensor ◽  
High Volume ◽  
Cauchy Stress ◽  
Commercial Software ◽  
Steel Cylinder ◽  
Kirchhoff Stress ◽  
Jaumann Rate ◽  
Objective Stress

The objective stress rates used in most commercial finite element programs are the Jaumann rate of Kirchhoff stress, Jaumann rates of Cauchy stress, or Green–Naghdi rate. The last two were long ago shown not to be associated by work with any finite strain tensor, and the first has often been combined with tangential moduli not associated by work. The error in energy conservation was thought to be negligible, but recently, several papers presented examples of structures with high volume compressibility or a high degree of orthotropy in which the use of commercial software with the Jaumann rate of Cauchy or Kirchhoff stress leads to major errors in energy conservation, on the order of 25–100%. The present paper focuses on the Green–Naghdi rate, which is used in the explicit nonlinear algorithms of commercial software, e.g., in subroutine VUMAT of ABAQUS. This rate can also lead to major violations of energy conservation (or work conjugacy)—not only because of high compressibility or pronounced orthotropy but also because of large material rotations. This fact is first demonstrated analytically. Then an example of a notched steel cylinder made of steel and undergoing compression with the formation of a plastic shear band is simulated numerically by subroutine VUMAT in ABAQUS. It is found that the energy conservation error of the Green–Naghdi rate exceeds 5% or 30% when the specimen shortens by 26% or 38%, respectively. Revisions in commercial software are needed but, even in their absence, correct results can be obtained with the existing software. To this end, the appropriate transformation of tangential moduli, to be implemented in the user's material subroutine, is derived.

Biaxial mechanics of excised canine pulmonary arteries

1995 ◽  
Vol 269 (2) ◽  
pp. H433-H442 ◽  
Author(s):  
J. C. Debes ◽  
Y. C. Fung
Keyword(s):  
Energy Function ◽  
Axial Strain ◽  
Testing Machine ◽  
Pulmonary Arteries ◽  
Strain Energy Function ◽  
Tensile Testing Machine ◽  
Physiological Level ◽  
Kirchhoff Stress ◽  
Biaxial Tests ◽  
Right Pulmonary Artery

A new method has been developed for measuring the stress-strain relationship in excised canine pulmonary arteries. Segments of dog main right pulmonary arteries were isolated by making two transverse cuts at each end of a segment near the bifurcations, yielding short cylinders, which were then cut radially, relieving the residual stress, causing the cylindrical shells to spring open to approximately flat rectangular slabs with dimensions approximately 1.0 x 3.0 x 0.1 cm. The specimens were then tested using a biaxial tensile testing machine. The resulting data show an approximately linear relationship between Kirchhoff stress and Lagrangian strain with very little hysteresis. The following pseudostrain energy function serves as a practical approximation for pulmonary arteries subjected to physiological levels of stress and strain: rho 0W(2) = 1/2(a1E2xx + a2E2yy + 2 a4ExxEyy), where rho 0 is the density of the wall (mass per unit volume), W is the energy per unit mass [superscript "(2)" indicates this is a 2-dimensional strain energy function], E is strain, a1, a2, and a4 are material constants with units of stress, and the subscripts x and y refer to the circumferential and axial axes, respectively, of the artery. To assess the physiological level of strain in the main right pulmonary artery, vessels were perfused in situ at physiological pressure (26 cmH2O) with silicone elastomer. The arteries were then excised and marked with small ink spots. Photographs of the spots on four tangent planes of the excised artery indicate a maximum circumferential strain of 21.5% and a maximum axial strain of 36.5% relative to the zero-stress state. These values are within the range of strain used in the biaxial tests. The relationship between Kirchhoff stress and Green's strain is approximately linear within the physiological range. The stress levels required to cause tissue failure are at least 10 times greater than the estimated normal physiological level.

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