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# Strain Components and Compatibility Equations

There are __________ strain components for a three dimensional case.

a) 2

b) 3

c) 6

d) 8

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Answer: c

Explanation: There are six strain components for a three dimensional case and there are three linear strain components and three shearing strain components.

If u is the displacement in x-direction, the linear strain component in is defined by __________

a) εX=∂u∂x

b) εX=∂x∂u

c) εX=dxdu

d) ε_{X}=x+u

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Answer: a

Explanation: The strain is the ratio of change in dimension of an object to its original dimension. Therefore, the linear strain component in is defined by,

εX=∂u∂x. Where, u is the displacement in x-direction.

If v is the displacement in y-direction, the linear strain component in is defined by __________

a) εY=∂v∂y

b) εY=∂y∂v

c) εY=dydv

d) ε_{Y}=v+y

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Answer: a

Explanation: The strain is the ratio of change in dimension of an object to its original dimension. Therefore, the linear strain component in is defined by,

εY=∂v∂y. Where, v is the displacement in y-direction.

If w is the displacement in z-direction, the linear strain component in is defined by __________

a) εZ=∂w∂z

b) εZ=∂z∂w

c) εZ=dzdw

d) ε_{Z}=z+w

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Answer: a

Explanation: The strain is the ratio of change in dimension of an object to its original dimension. Therefore, the linear strain component in is defined by,

εZ=∂w∂z. Where, w is the displacement in z-direction.

The shearing strain component in X-Y plane is ________

a) Γxy=∂v∂x+∂u∂y

b) Γxy=∂v∂x−∂u∂y

c) Γxy=∂v∂y+∂u∂x

d) Γxy=∂v∂y−∂u∂x

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Answer: a

Explanation: In order to find the shearing strain component in X-Y plane, let us consider a plane lamina of size dx, dy in X-Y plane. If u is the displacement in x-direction and v is the displacement in y-direction, the shearing strain component is,

Γxy=∂v∂x+∂u∂y.

The shearing strain component in Y-Z plane is ________

a) Γyz=∂w∂y+∂v∂z

b) Γyz=∂v∂x−∂w∂y

c) Γyz=∂v∂y+∂w∂x

d) Γyz=∂v∂y−∂w∂x

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Answer: a

Explanation: In order to find the shearing strain component in Y-Z plane, let us consider a plane lamina of size dy, dz in Y-Z plane. If w is the displacement in z-direction and v is the displacement in y-direction, the shearing strain component is,

Γyz=∂w∂y+∂v∂z.

The shearing strain component in Z-X plane is ________

a) Γxz=∂w∂y+∂v∂z

b) Γxz=∂v∂x−∂w∂y

c) Γxz=∂u∂z+∂w∂x

d) Γxz=∂v∂y−∂w∂x

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Answer: c

Explanation: In order to find the shearing strain component in Z-X plane, let us consider a plane lamina of size dx, dz in Z-X plane. If w is the displacement in z-direction and u is the displacement in x-direction, the shearing strain component is,

Γxz=∂u∂z+∂w∂x.

The strain tensor is given by ____________

a) ⎛⎝⎜εxzεyxεzxεxyεyzεzyεxxεyyεzz⎞⎠⎟

b) ⎛⎝⎜εxyεyxεzyεxxεyyεzzεxzεyzεzz⎞⎠⎟

c) ⎛⎝⎜εxxεyxεzxεxyεyyεzyεxzεyzεzz⎞⎠⎟

d) ⎛⎝⎜εxxεyyεzzεxyεyxεzyεxzεyzεzx⎞⎠⎟

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Answer: c

Explanation: The strain tensor is given by,

⎛⎝⎜εxxεyxεzxεxyεyyεzyεxzεyzεzz⎞⎠⎟

where, the diagonal elements represent the linear strain components and the off-diagonal elements represent the shearing strain components.

The strain tensor is given by ____________

a) ⎛⎝⎜1/2Γxz1/2Γyx1/2Γzx1/2Γxy1/2Γyz1/2Γzyεxxεyyεzz⎞⎠⎟

b)⎛⎝⎜1/2Γxy1/2Γyx1/2Γzyεxxεyyεzz1/2Γxz1/2Γyz1/2Γzz⎞⎠⎟

c) ⎛⎝⎜εxx1/2Γyx1/2Γzx1/2Γxyεyy1/2Γzy1/2Γxz1/2Γyzεzz⎞⎠⎟

d) ⎛⎝⎜εxxεyyεzz1/2Γxy1/2Γyx1/2Γzy1/2Γxz1/2Γyz1/2Γzx⎞⎠⎟

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Answer: c

Explanation: The strain tensor is given by,

⎛⎝⎜εxxεyxεzxεxyεyyεzyεxzεyzεzz⎞⎠⎟

where, the diagonal elements represent the linear strain components and the off-diagonal elements represent the shearing strain components. The linear strain components of a diagonal is equal to half the sum shearing strain components.

∴ ε_{yx}=1/2Γ_{yx}

ε_{xz}=1/2Γ_{xz}

ε_{zy}=1/2Γ_{zy}.

∴ the strain tensor is given by,

⎛⎝⎜εxx1/2Γyx1/2Γzx1/2Γxyεyy1/2Γzy1/2Γxz1/2Γyzεzz⎞⎠⎟.

The linear strain of a diagonal is equal to half the sum shearing strain components.

a) True

b) False

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Answer: a

Explanation: The linear strain of a diagonal is equal to half the sum shearing strain components. Thus if ε_{yx}, ε_{xz} and ε_{zy} represent the linear strains of the diagonal of a plane lamina,

∴ ε_{yx}=1/2Γ_{yx}

ε_{xz}=1/2Γ_{xz}

ε_{zy}=1/2Γ_{zy.}

The compatibility equations are results of application of stress equations.

a) True

b) False

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Answer: b

Explanation: The equations resulting from the applications of the strain equations are known as the compatibility equations or the Saint-Venant’s equations.

___________ is a compatibility equation.

a) ∂2εy∂y2+∂2εy∂x2=∂2Γxy∂x∂y

b) ∂2εx∂y2+∂2εy∂x2=∂2Γxy∂x∂y

c) ∂2εx∂y2+∂2εy∂x2=∂2Γxy∂x∂y

d) ∂2εx∂y2+∂2εy∂x2=∂2Γxy∂x∂y

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Answer: b

Explanation: The strain equations in terms of displacements is given by,

εX=∂u∂x—————−(1)

εY=∂v∂y—————−(2)

Γxy=∂v∂x+∂u∂y———(3)

Differentiating (1) twice with respect to y, (2) twice with respect to x and (3) once with respect to x and then y,

∂2εx∂y2=∂3udx∂y2————−(4)

∂2εy∂x2=∂3vdy∂x2————−(5)

∂2Γxy∂x∂y=∂3udx∂y2+∂3vdy∂x2——−(6)

∴ from (4), (5) and (6), we get,

∂2εx∂y2+∂2εy∂x2=∂2Γxy∂x∂y which is a compatibility equation.