Applications of a Rotation-Free Triangular Element for Finite Strain Analysis of Thin Shells and Membranes

Possible restrictions on isotropic constitutive laws for finitely deformed elastic solids are examined from the standpoint of Hill (1968). This introduced the notion of conjugate pairs of stress and strain measures, whereby families of contending inequalities can be generated. A typical member inequality stipulates that the scalar product of the rates of change of certain conjugate variables is positive in all circumstances. Interrelations between the various inequalities are explored, and some statical implications are established. The discussion depends on several ancillary theorems which are apparently new; these have, in addition, an intrinsic interest in the broad field of basic stress—strain analysis.

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Finite Strain Analysis Using Shape and Point Data

Developments in Structural Geology and Tectonics - Problems and Solutions in Structural Geology and Tectonics ◽

10.1016/b978-0-12-814048-2.00008-9 ◽

2019 ◽

pp. 89-104

Author(s):

Kieran F. Mulchrone

Keyword(s):

Finite Strain ◽

Strain Analysis ◽

Point Data

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The Elastic Properties of β-Mg2SiO4 Containing 0.73 wt.% of H2O to 10 GPa and 600 K by Ultrasonic Interferometry with Synchrotron X-Radiation

Minerals ◽

10.3390/min10030209 ◽

2020 ◽

Vol 10 (3) ◽

pp. 209

Author(s):

Gabriel D. Gwanmesia ◽

Matthew L. Whitaker ◽

Lidong Dai ◽

Alwin James ◽

Haiyan Chen ◽

...

Keyword(s):

Finite Strain ◽

Seismic Velocity ◽

Strain Analysis ◽

Data Sets ◽

Third Order ◽

Data Set ◽

Earth’S Mantle ◽

Ultrasonic Interferometry ◽

Linear Fit ◽

Derivatives Of

We measured the elastic velocities of a synthetic polycrystalline β-Mg2SiO4 containing 0.73 wt.% H2O to 10 GPa and 600 K using ultrasonic interferometry combined with synchrotron X-radiation. Third-order Eulerian finite strain analysis of the high P and T data set yielded Kso = 161.5(2) GPa, Go = 101.6(1) GPa, and (∂Ks/∂P)T = 4.84(4), (∂G/∂P)T = 1.68(2) indistinguishable from Kso = 161.1(3) GPa, Go = 101.4(1) GPa, and (∂Ks/∂P)T = 4.93(4), (∂G/∂P)T = 1.73(2) from the linear fit. The hydration of the wadsleyite by 0.73 wt.% decreases Ks and G moduli by 5.3% and 8.6%, respectively, but no measurable effect was noted for (∂Ks/∂P)T and (∂G/∂P)T. The temperature derivatives of the Ks and G moduli from the finite strain analysis (∂KS/∂T)P = −0.013(2) GPaK−1, (∂G/∂T)P = −0.015(0.4) GPaK−1, and the linear fit (∂KS/∂T)P = −0.015(1) GPaK−1, (∂G/∂T)P = −0.016(1) GPaK−1 are in agreement, and both data sets indicating the |(∂G/∂T)P| to be greater than |(∂KS/∂T)P|. Calculations yield ∆Vp(α-β) = 9.88% and ∆VS(α-β) = 8.70% for the hydrous β-Mg2SiO4 and hydrous α-Mg2SiO4, implying 46–52% olivine volume content in the Earth’s mantle to satisfy the seismic velocity contrast ∆Vs = ∆VP = 4.6% at the 410 km depth.

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Analysis of strained sedimentary fabrics: review and tests

Canadian Journal of Earth Sciences ◽

10.1139/e87-046 ◽

1987 ◽

Vol 24 (3) ◽

pp. 442-455 ◽

Cited By ~ 16

Author(s):

G. J. Borradaile

Keyword(s):

Finite Strain ◽

Strain Analysis ◽

Orientation Distribution ◽

Combined Effects ◽

Accurate Analysis ◽

Accurate Definition ◽

The Matrix ◽

Text Model ◽

Definition Of ◽

Homogeneous Strain

Accurate analysis of the homogeneous strain of ellipsoidal objects requires that certain premises are met. These are (1) coaxiality of strain increments, (2) accurate definition of finite strain axes, and (3) perfectly passive behaviour of the objects with respect to the matrix. The [Formula: see text] equations provide a mathematical model for this idealized scenario.In practice the analysis is limited to data from plane surfaces, and its success depends upon the type of initial fabric of the objects, the frequency distribution of their initial shapes, and errors of measurement of the final shapes (Rf1/2) and of final orientations [Formula: see text]. Together with the difficulty in locating the finite strain axes, these factors greatly affect the outcome of the analysis, and their combined effects have not been studied previously.Detailed observations on the orientations of strain and of the orientation–distribution of the objects' long axes sometimes indicate that the [Formula: see text] model is incompatible with the data. Strain analysis is then not possible by the present methods. If the data are compatible with the [Formula: see text] model and if the original fabric can be shown to have been random or had a preferred orientation parallel to bedding, it is possible to determine the strain accurately to within a few percent. This has been partly confirmed in practical examples in which the strain has been independently determined using the tectonic distortion of rims of individual clasts.

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