StressStrain-Review

# StressStrain-Review - zz Stress Strain A review zx zy yz xx...

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1 zz Stress & Strain: A review xx yy yz xy xz zx zy yy xx zz yx 1 of 79 Erik Eberhardt – UBC Geological Engineering EOSC 433 Disclaimer before beginning your problem assignment: Pick up and compare any set of textbooks on rock mechanics, soil mechanics or solid mechanics, and you will find that the discussion on Mohr Circles, stress-strain analysis, matrix math, etc., either uses different conventions or contains a typo that will throw your calculations off. Clockwise is positive, clockwise is negative, mathematical shear strain, engineering shear strain… It all seems rather confusing. But instead of becoming frustrated or condemning the proof-reader of a given textbook (or these notes), I like to look at it as a good lesson in not relying 100% on something, especially at the expense of your judgement. The notes that follow come from several sources and I have tried to eliminate the errors when I find them. However, when using these notes to complete your problem assignment, try to also use your judgement as to 2 of 79 Erik Eberhardt – UBC Geological Engineering EOSC 433 whether the answer you obtain makes sense. If not, consult a different source to double check to see if there was an error. On that note, if you find an error and/or a source that you would recommend as having given you a clearer understanding of a particular calculation, please let me know.

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2 Understanding Stress There is a fundamental difference both conceptually and Stress is not familiar: it is a tensor quantity and tensors are not encountered in everyday life. There is a fundamental difference, both conceptually and mathematically, between a tensor and the more familiar quantities of scalars and vectors: Scalar: a quantity with magnitude only (e.g. temperature, time, mass). Vector: a quantity with magnitude and direction (e.g. force, velocity, acceleration). 3 of 79 Erik Eberhardt – UBC Geological Engineering EOSC 433 Tensor: a quantity with magnitude and direction, and with reference to a plane it is acting across (e.g. stress, strain, permeability). Both mathematical and engineering mistakes are easily made if this crucial difference is not recognized and understood. The Stress Tensor The second-order tensor which we will be examining has: - 9 components of which 6 are independent; - values which are point properties; - values which depend on orientation relative to a set of reference axes; - 6 of the 9 components becoming zero at a particular orientation; 4 of 79 Erik Eberhardt – UBC Geological Engineering EOSC 433 - three principal components; - complex data reduction requirements because two or more tensors cannot, in general, be averaged by averaging the respective principal stresses.
3 Components of Stress On a real or imaginary plane through a material, there can be normal forces and shear forces . These forces create the stress tensor. The normal and shear stress components are the normal and shear forces per unit area.

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StressStrain-Review - zz Stress Strain A review zx zy yz xx...

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