1) If the deformation mapping of a body is given by x(X)=(X, + AX, )e, +(X, + BX, )e, + X3zez where A and B are constants. Determine (a) the displacement components in the material description (b) the displacement components in the spatial description (c) the components of the Green strain tensor (d) the components of the small strain tensor under the condition of small displacement gradients

1) If the deformation mapping of a body is given by x(X)=(X, + AX, )e, +(X, + BX, )e, + X3zez where A and B are constants. Determine (a) the displacement components in the material description (b) the displacement components in the spatial description (c) the components of the Green strain tensor (d) the components of the small strain tensor under the condition of small displacement gradients

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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3) The displacement field of a screw dislocation of Burger's vector b is given by u = v = 0, W = tan -1y 2n Find the Cartesian and polar components of the strain tensor.
(Consider the stress tensor in a given point of a body): 2 1 Oij 2 0 2 1 2 (Determine o rag and identify the plane (the versor of its normal) for which the stress vector is a null vector.)
what is: 1.The deformation mapping and components of the deformation gradient 2.The components of the Cauchy stress tensor and the first PioIa-Kirchhoff stress tensor. 3.Under what conditions the two tensors are equal
X₁ = X₁ + 2X3, x2 = X₂ - 2X3, X3 = −2X₁ + 2X2 + X3 Consider the two line elements dX(¹) = ₁, dX(²) = ₂ emanating from the the origin (0, 0,0). Use the Right Cauchy Green strain tensor to determine whether these elements in the current configuration (dr(¹), dx(²)) are perpendicular. Use the Right Cauchy-Green strain tensor to evaluate the stretch of the line ele- ment dX = ₁ +ē2, and hence determine whether the element contract, stretches, or stays the same lenght after deformation. Determine the Green-Lagrangian and Eulerian strain tensors. Decompose the deformation into a stretching and rotation tensors. Describe what kind of deformation the body is undergoing by including what the prin- cipal stretches are and the type of rotation the body is undergoing.
. Determine the displacement gradient tensor, right Cauchy-Green deformation tensor, small deformations strain tensor, rotation tensor, and small deformations and rotations strain tensor for the following displacement fields: u={A X₁² X3, B X1 X2 X3,, C (X2 + X32)} where A, B, and C are constants.
The 3-dimensional state of strain at a material point in x, y, z Q4 coordinates is given by: E ya [ε]=Ey Eyy - M E E 20 E יה = -120 0-150 0 0 0 -150 0 240 *10-6 (a) Calculate the volumetric strain and the deviatoric strain invariants (b) Calculate the mean stress and the deviatoric stress tensor (c) Write the characteristic equation of strain (d) Write the characteristic equation of stress The material is linear elastic (E=210GPa, v=0.3).
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Imagine a 3D state of strain in a mechanical part, measured perhaps by digital volume correlation or other means, given in Cartesian coordinates by €rx = a(x² + y²), €yy = a(y² +2²), Exy=b(xyz), Exz = €yz = €zz = 0, where a and b are constants and x, y, z are coordinates in space. Determine whether this measured strain field is a possible state of strain for a continuum that remains intact and continuous after deformation.
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