Rotation and strain of linear and planar structures in three-dimensional progressive deformation

Rotation and progressive strain have been studied for a sheet embedded in a matrix which undergoes rotational three-dimensional strain under constant volume conditions. The mathematics gives explicit information on the following features:

1. (1) The length and position (relative to a defined coordinate system) of the principal axes of the strain ellipsoid at any stage of the progressive deformation.

2. (2) The position and length of the principal axes in any plane intersecting the strain ellipsoid, also at any stage of the deformation.

3. (3) The position and length of passive markers which initially coincided with the principal axes in an intersecting plane. This is of consequence for the distinction between passively rotating structures and actively forming structures.

4. (4) The shear strain parallel to an intersecting plane or sheet, as indicated by the angular difference between the normal to an intersecting plane at any time and the marker at the same time which initially, however, was parallel to the normal. This layer-parallel shear causes boudins to rotate and the axial plane of buckles to tilt.

The relationships have been expressed quantitatively in the bulk of the paper and illustrated in diagrams. The analysis presented is basic for the study of the deformational behavior of competent sheets of rocks embedded in less competent ones.     

Lodgement till and flow till: a discussion

A comment by J. Kriiger on I. Marcussen's paper 'Distinguishing between lodgement till and flow till in Weichselian deposits' (Boreas 4 , pp. 113–123) is followed by a reply from I. Marcussen.     

Particle paths, displacement and progressive strain applicable to rocks

If the particle paths are known for deforming continuous media such as rocks, the strain is determined at all stages of the deformation. The particle paths are studied for various types of simultaneous combinations of pure shear and simple shear. Any kind of progressive plane homogeneous strain can be expressed as a simultaneous superposition of pure shear and simple shear by selecting the proper ratio between the two strain rates and the proper angle between the slide direction of the simple-shear part and the principal axes of the pure-shear part. In the cases studied — except one — the angle between the slide direction of the simple-shear part and the principal strain rate of the pure-shear part is 45°. Several combinations of the simple-shear rate, γ, and the pure-shear rate, , are tested. These combinations give particle paths varying from sets of straight parallel lines to orthogonal hyperbolas. Distorted hyperbolas, ellipses and circles constitute the particle paths at intermediate ratios. From the particle-path equations — which are found by integration of the rate-of-deformation equations — the strain ellipse is readily determined at any stage of the deformation. One particularly intriguing result is the rotating and pulsating strain ellipse found in the cases when the particle paths are closed curves (ellipses). Application of the results to various fold-, thrust- and inclusion structures is suggested. In an appendix the treatment of rotational deformation as a superposition of irrttational strain and rigid rotation is considered for comparison.     

A simple method of quantifying filamentous algae in microscope

A fast and simple method to quantify filamentous algae in inverted microscope is described. It is based on the number of intersections between the midline of a diametrical transect and the filaments on the chamber bottom. It is shown to be theoretically sound and is compared with the common length measurement method.     

Grazing by a colourless flagellate on a green algal bloom

A dense bloom of a green algae was reduced by 97% within 5 days. This was probably caused by grazing by a colourless flagellate that increased logarithmic to high number under these days and disappeared as quickly. After this the green algae increased rapidly and mucilage lumps excreated by the flagellate was colonized by a specialistChlomydomonas. The flagellate attack was probably enhanced by the high cell density and the physiological weakness of the green algae, due to a very low growth caused by shortage of inorganic carbon. The grazing seems to have caused an increase of inorganic nutrients that promoted the following growth of the green algae.     

Contact strain and folding instability of a multilayered body under compression

The contact strain which develops in a host rock during compression-folding of a more competent layer is treated mathematically and demonstrated by tests on composite models of rubber sheets and blocks with different rigidities.The contact strain is found to be negligible outside a zone about one initial wavelength wide on either side of the folded sheet. This theoretical conclusion based upon the contact deformation resulting from the initial small-amplitude buckles is also, according to the experimental tests, valid for rather strong buckling.Second-order folding of a thin competent layer laying within the contact zone of a thicker buckled competent layer is demonstrated by experimental tests.Considerations of the stresses connected with the contact strain and the stress distribution in the buckling competent layer itself give a simple expression for the initial stable wavelength which develops in response to compression parallel to the layering.The strain distribution in folded multilayers consisting of a set of thin competent sheets evenly spaced in an uniform less competent body is considered theoretically.In such multilayers the stable initial wavelength as developed during compression will, according to the theory, increase somewhat with decreasing spacing between the competent layers if all other conditions are equal. If the spacing is close enough (the limit depends upon the relative mechanical properties of the two substances) the multilayer will buckle as an unit in the sense that one half-wave covers the whole length of the body. This agrees with compression tests on multilayered rubber blocks.In the last two sections of the paper a theory of viscous buckling of thick layers is developed both for the case of a single layer enclosed in a large uniform body and for the case of a multilayer of alternating competent and incompetent layers. In this theory shear parallel to layering plays an important part. For a single embedded layer the thick-layer model gives a little smaller wavelength/thickness ratio than the simple but less realistic thin-layer model. For multilayers strain parallel to layering becomes very significant, particularly when the spacing between the competent layers is small, and the thick-layer model differs considerably from the simpler thin-layer model.A theory of buckling of a multilayer withn alternating competent-incompetent layers sandwiched between two very thick uniform bodies is also developed.