I am attaching the solution. Please explain the derivation in your handwritten form. Dont type please Derive the following relationship between the strain, sij(x, y, z) and the displacement field u(x, y, z)Eij=JuxӘxaux1/диу, дихдх ду+21(duz ux)+2дх дz1/дих2ду+NIHдиу?xduyду1/duz uy2 ду дz+1/дих д+2дz ?x1/ uу, дuz+2 дzдuzдzду) Figure : Example of displacement vector field for a solid anchored at the bottom and with a force (left toright) applied on the top. All other solid surfaces (but the bottom) can move freely. Strains are a function ofthe displacement field.De formedF UndeformedYet, absolute displacements are not enough to determine stresses. A solid may translate or rotate in space withoutdevelopment of any internal stresses required to equilibrate external actions (imagine a cookie "floating" in zero gravitywithin the International Space StationLet'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 achange of volume of the solid and therefore contributes to volumetric strain. The resulting deformation or strain(change of length divided original length) is£=↑Displacement UfieldX₂18112• (CENTER) A solid is stretched on its face 2 in direction 2 only. This type of deformation also contributes tovolumetric strain. The resulting strain is1/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,Ax₁duxдгуE11=. (RIGHT) The solid is now distorted. Notice that the faces do not make a right angle anymore. The change of volumeis negligible for small deformations. The resulting distortion or shear strain is proportional to the change in anglebetween faces 1 and 2. Hence the change of angle is - [- arctan(Au₁/A₂)+ arctan(Au₂/A1)]. The shearstrain is 1/2 of the total change of the angle and therefore (for small changes arctan(z/y) ~ 1/y)€12=X₂€22=AutAri822duz8x21/?uz ?u12dr8r₂+?us ?ui1 duz duz++Ori Ors 28х2 т Әхз1/8u₁ duz2 0₂ 01+Au₂Ar₂AllAx₂AU₂AX₂The summation of all diagonal terms yields the volumetric straindus1/?u20rs 01Ax₂1/duz მus2 813 81₂+Jusdrsx, του,Au,Strains do not quantify the absolute value of displacements, but its variation in space (derivative with respect to z;). Allother strains are found with similar equations in the 3D case. Similarly to the stress tensor, strains can be organized in atensor where elements in the diagonal contribute to volumetric strain, and off-diagonal elements are shear strains.Evol 11 +22+ €33Strains (EExtensionX₂$12.Contraction↓=Ax₂1A+Au Au2 Ax₂ Ax,.AU₂(3.4)(3.5)E11 12 1321 22 23E31 E32 E33(3.6)(3.8)

Question

I am attaching the solution. Please explain the derivation in your handwritten form. Dont type please Derive the following relationship between the strain, sij(x, y, z) and the displacement field u(x, y, z)Eij=JuxӘxaux1/диу, дихдх ду+21(duz ux)+2дх дz1/дих2ду+NIHдиу?xduyду1/duz uy2 ду дz+1/дих д+2дz ?x1/ uу, дuz+2 дzдuzдzду) Figure : Example of displacement vector field for a solid anchored at the bottom and with a force (left toright) applied on the top. All other solid surfaces (but the bottom) can move freely. Strains are a function ofthe displacement field.De formedF UndeformedYet, absolute displacements are not enough to determine stresses. A solid may translate or rotate in space withoutdevelopment of any internal stresses required to equilibrate external actions (imagine a cookie &#34;floating&#34; in zero gravitywithin the International Space StationLet'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 achange of volume of the solid and therefore contributes to volumetric strain. The resulting deformation or strain(change of length divided original length) is£=↑Displacement UfieldX₂18112• (CENTER) A solid is stretched on its face 2 in direction 2 only. This type of deformation also contributes tovolumetric strain. The resulting strain is1/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,Ax₁duxдгуE11=. (RIGHT) The solid is now distorted. Notice that the faces do not make a right angle anymore. The change of volumeis negligible for small deformations. The resulting distortion or shear strain is proportional to the change in anglebetween faces 1 and 2. Hence the change of angle is - [- arctan(Au₁/A₂)+ arctan(Au₂/A1)]. The shearstrain is 1/2 of the total change of the angle and therefore (for small changes arctan(z/y) ~ 1/y)€12=X₂€22=AutAri822duz8x21/?uz ?u12dr8r₂+?us ?ui1 duz duz++Ori Ors 28х2 т Әхз1/8u₁ duz2 0₂ 01+Au₂Ar₂AllAx₂AU₂AX₂The summation of all diagonal terms yields the volumetric straindus1/?u20rs 01Ax₂1/duz მus2 813 81₂+Jusdrsx, του,Au,Strains do not quantify the absolute value of displacements, but its variation in space (derivative with respect to z;). Allother strains are found with similar equations in the 3D case. Similarly to the stress tensor, strains can be organized in atensor where elements in the diagonal contribute to volumetric strain, and off-diagonal elements are shear strains.Evol 11 +22+ €33Strains (EExtensionX₂$12.Contraction↓=Ax₂1A+Au Au2 Ax₂ Ax,.AU₂(3.4)(3.5)E11 12 1321 22 23E31 E32 E33(3.6)(3.8)

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