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Derive the following relationship between the strain, sij(x, y, z) and the displacement field u(x, y, z) Eij дих дх 1/диу дих 2дх + ду 1/ uz дих 2дх + дz 12 12 1/2 (Dux + dux) / (0ux + 1/дих + ду July ду ди, диу + 2 ду дz 1/диz 12 12 дих ди, Әх 1/диу дuz\ + ду дцz дz

Derive the following relationship between the strain, sij(x, y, z) and the displacement field u(x, y, z) Eij дих дх 1/диу дих 2дх + ду 1/ uz дих 2дх + дz 12 12 1/2 (Dux + dux) / (0ux + 1/дих + ду July ду ди, диу + 2 ду дz 1/диz 12 12 дих ди, Әх 1/диу дuz\ + ду дцz дz

Elements Of Electromagnetics
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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Derive the following relationship between the strain, sij(x, y, z) and the displacement field u(x, y, z) Eij = Jux Әx aux 1/диу, дих дх ду + 2 1(duz ux) + 2дх дz 1/дих 2 ду + NIH диу ?x duy ду 1/duz uy 2 ду дz + 1/дих д + 2дz ?x 1/ uу, дuz + 2 дz дuz дz ду )
Transcribed Image Text:Derive the following relationship between the strain, sij(x, y, z) and the displacement field u(x, y, z) Eij = Jux Әx aux 1/диу, дих дх ду + 2 1(duz ux) + 2дх дz 1/дих 2 ду + NIH диу ?x duy ду 1/duz uy 2 ду дz + 1/дих д + 2дz ?x 1/ uу, дuz + 2 дz дuz дz ду )
Figure : Example of displacement vector field for a solid anchored at the bottom and with a force (left to right) applied on the top. All other solid surfaces (but the bottom) can move freely. Strains are a function of the displacement field. De formed F Undeformed Yet, absolute displacements are not enough to determine stresses. A solid may translate or rotate in space without development of any internal stresses required to equilibrate external actions (imagine a cookie "floating" in zero gravity within the International Space Station Let's look at Figure in order to relate displacements to strains: • (LEFT) A solid is stretched on its face 1 (perpendicular to e₁) in direction 1 only. This type of deformation produces a change of volume of the solid and therefore contributes to volumetric strain. The resulting deformation or strain (change of length divided original length) is £= ↑ Displacement U field X₂ 1 811 2 • (CENTER) A solid is stretched on its face 2 in direction 2 only. This type of deformation also contributes to volumetric strain. The resulting strain is 1/Au, Au2) 2 AT AT1/ The average in the equation ensures capturing shear distortion rather than rotation. Figure 3.8: Strain equations for small deformations. Ax₂ Au₂ ** Au, Ax₁ dux дгу E11= . (RIGHT) The solid is now distorted. Notice that the faces do not make a right angle anymore. The change of volume is negligible for small deformations. The resulting distortion or shear strain is proportional to the change in angle between faces 1 and 2. Hence the change of angle is - [- arctan(Au₁/A₂)+ arctan(Au₂/A1)]. The shear strain is 1/2 of the total change of the angle and therefore (for small changes arctan(z/y) ~ 1/y) €12= X₂ €22= Aut Ari 822 duz 8x2 1/?uz ?u1 2dr8r₂ + ?us ?ui 1 duz duz + + Ori Ors 28х2 т Әхз 1/8u₁ duz 2 0₂ 01 + Au₂ Ar₂ All Ax₂ AU₂ AX₂ The summation of all diagonal terms yields the volumetric strain dus 1/?u 20rs 01 Ax₂ 1/duz მus 2 813 81₂ + Jus drs x, του, Au, Strains do not quantify the absolute value of displacements, but its variation in space (derivative with respect to z;). All other strains are found with similar equations in the 3D case. Similarly to the stress tensor, strains can be organized in a tensor where elements in the diagonal contribute to volumetric strain, and off-diagonal elements are shear strains. Evol 11 +22+ €33 Strains (E Extension X₂ $12. Contraction ↓ = Ax₂ 1 A+ Au Au 2 Ax₂ Ax,. AU₂ (3.4) (3.5) E11 12 13 21 22 23 E31 E32 E33 (3.6) (3.8)
Transcribed Image Text:Figure : Example of displacement vector field for a solid anchored at the bottom and with a force (left to right) applied on the top. All other solid surfaces (but the bottom) can move freely. Strains are a function of the displacement field. De formed F Undeformed Yet, absolute displacements are not enough to determine stresses. A solid may translate or rotate in space without development of any internal stresses required to equilibrate external actions (imagine a cookie "floating" in zero gravity within the International Space Station Let's look at Figure in order to relate displacements to strains: • (LEFT) A solid is stretched on its face 1 (perpendicular to e₁) in direction 1 only. This type of deformation produces a change of volume of the solid and therefore contributes to volumetric strain. The resulting deformation or strain (change of length divided original length) is £= ↑ Displacement U field X₂ 1 811 2 • (CENTER) A solid is stretched on its face 2 in direction 2 only. This type of deformation also contributes to volumetric strain. The resulting strain is 1/Au, Au2) 2 AT AT1/ The average in the equation ensures capturing shear distortion rather than rotation. Figure 3.8: Strain equations for small deformations. Ax₂ Au₂ ** Au, Ax₁ dux дгу E11= . (RIGHT) The solid is now distorted. Notice that the faces do not make a right angle anymore. The change of volume is negligible for small deformations. The resulting distortion or shear strain is proportional to the change in angle between faces 1 and 2. Hence the change of angle is - [- arctan(Au₁/A₂)+ arctan(Au₂/A1)]. The shear strain is 1/2 of the total change of the angle and therefore (for small changes arctan(z/y) ~ 1/y) €12= X₂ €22= Aut Ari 822 duz 8x2 1/?uz ?u1 2dr8r₂ + ?us ?ui 1 duz duz + + Ori Ors 28х2 т Әхз 1/8u₁ duz 2 0₂ 01 + Au₂ Ar₂ All Ax₂ AU₂ AX₂ The summation of all diagonal terms yields the volumetric strain dus 1/?u 20rs 01 Ax₂ 1/duz მus 2 813 81₂ + Jus drs x, του, Au, Strains do not quantify the absolute value of displacements, but its variation in space (derivative with respect to z;). All other strains are found with similar equations in the 3D case. Similarly to the stress tensor, strains can be organized in a tensor where elements in the diagonal contribute to volumetric strain, and off-diagonal elements are shear strains. Evol 11 +22+ €33 Strains (E Extension X₂ $12. Contraction ↓ = Ax₂ 1 A+ Au Au 2 Ax₂ Ax,. AU₂ (3.4) (3.5) E11 12 13 21 22 23 E31 E32 E33 (3.6) (3.8)
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