Displacement

In physics, **deformation** is the continuum mechanics transformation of a body from a *reference* configuration to a *current* configuration. A configuration is a set containing the positions of all particles of the body.

A deformation may be caused by external loads, body forces (such as gravity or electromagnetic forces), or changes in temperature, moisture content, or chemical reactions, etc.

**Strain** is a description of deformation in terms of *relative* displacement of particles in the body that excludes rigid-body motions. Different equivalent choices may be made for the expression of a strain field depending on whether it is defined with respect to the initial or the final configuration of the body and on whether the metric tensor or its dual is considered.

In a continuous body, a deformation field results from a stress field induced by applied forces or is due to changes in the temperature field inside the body. The relation between stresses and induced strains is expressed by constitutive equations, e.g., Hooke's law for linear elastic materials. Deformations which are recovered after the stress field has been removed are called **elastic deformations**. In this case, the continuum completely recovers its original configuration. On the other hand, irreversible deformations remain even after stresses have been removed. One type of irreversible deformation is **plastic deformation**, which occurs in material bodies after stresses have attained a certain threshold value known as the *elastic limit* or yield stress, and are the result of slip, or dislocation mechanisms at the atomic level. Another type of irreversible deformation is **viscous deformation**, which is the irreversible part of viscoelastic deformation.

In the case of elastic deformations, the response function linking strain to the deforming stress is the compliance tensor of the material.

## Displacement

A change in the configuration of a continuum body results in a displacement. The displacement of a body has two components: a rigid-body displacement and a deformation. A rigid-body displacement consists of a simultaneous translation and rotation of the body without changing its shape or size. Deformation implies the change in shape and/or size of the body from an initial or undeformed configuration *κ*0(**B**) to a current or deformed configuration *κt*(**B**) (Figure 1).

If after a displacement of the continuum there is a relative displacement between particles, a deformation has occurred. On the other hand, if after displacement of the continuum the relative displacement between particles in the current configuration is zero, then there is no deformation and a rigid-body displacement is said to have occurred.

The vector joining the positions of a particle *P* in the undeformed configuration and deformed configuration is called the displacement vector **u**(**X**,*t*) = *u**i***e***i* in the Lagrangian description, or **U**(**x**,*t*) = *U**J***E***J* in the Eulerian description.

A *displacement field* is a vector field of all displacement vectors for all particles in the body, which relates the deformed configuration with the undeformed configuration. It is convenient to do the analysis of deformation or motion of a continuum body in terms of the displacement field. In general, the displacement field is expressed in terms of the material coordinates as

u

(

X

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t
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b

(

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x

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−

X

or

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α

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b

J

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X

J

{\displaystyle \ \mathbf {u} (\mathbf {X} ,t)=\mathbf {b} (\mathbf {X} ,t)+\mathbf {x} (\mathbf {X} ,t)-\mathbf {X} \qquad {\text{or}}\qquad u\_{i}=\alpha \_{iJ}b\_{J}+x\_{i}-\alpha \_{iJ}X\_{J}}

or in terms of the spatial coordinates as

U

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x

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t
# )

b

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x

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)

x

−

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)

or

U

J

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α

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i

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J

{\displaystyle \ \mathbf {U} (\mathbf {x} ,t)=\mathbf {b} (\mathbf {x} ,t)+\mathbf {x} -\mathbf {X} (\mathbf {x} ,t)\qquad {\text{or}}\qquad U\_{J}=b\_{J}+\alpha \_{Ji}x\_{i}-X\_{J}\,}

where *αJi* are the direction cosines between the material and spatial coordinate systems with unit vectors **E***J* and **e***i*, respectively. Thus

E

J

⋅

e

i

=

α

J
i

=

α

i
J

{\displaystyle \ \mathbf E\_J \cdot \mathbf e\_i = \alpha\_{Ji}=\alpha\_{iJ}}

and the relationship between *ui* and *UJ* is then given by

u
i

=

α

i
J

U
J

or

U
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=

α

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i

u
i

{\displaystyle \ u\_i=\alpha\_{iJ}U\_J \qquad \text{or} \qquad U\_J=\alpha\_{Ji}u\_i}

Knowing that

e

i

=

α

i
J

E

J

{\displaystyle \ \mathbf e\_i = \alpha\_{iJ}\mathbf E\_J}

then

u

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X

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t
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u
i

e

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=

u
i

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α

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E

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U
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E

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U

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{\displaystyle \mathbf u(\mathbf X,t)=u\_i\mathbf e\_i=u\_i(\alpha\_{iJ}\mathbf E\_J)=U\_J\mathbf E\_J=\mathbf U(\mathbf x,t)}

It is common to superimpose the coordinate systems for the undeformed and deformed configurations, which results in **b** = 0, and the direction cosines become Kronecker deltas:

E

J

⋅

e

i

=

δ

J
i

=

δ

i
J

{\displaystyle \ \mathbf E\_J \cdot \mathbf e\_i = \delta\_{Ji}=\delta\_{iJ}}

Thus, we have

u

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X

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t
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x

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t
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−

X

or

u

i

=

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δ

i
J

X

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=

x

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i

{\displaystyle \ \mathbf {u} (\mathbf {X} ,t)=\mathbf {x} (\mathbf {X} ,t)-\mathbf {X} \qquad {\text{or}}\qquad u\_{i}=x\_{i}-\delta \_{iJ}X\_{J}=x\_{i}-X\_{i}}

or in terms of the spatial coordinates as

U

(

x

,
t
# )

x

−

X

(

x

,
t
)

or

U

J

=

δ

J
i

x

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=

x

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−

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J

{\displaystyle \ \mathbf {U} (\mathbf {x} ,t)=\mathbf {x} -\mathbf {X} (\mathbf {x} ,t)\qquad {\text{or}}\qquad U\_{J}=\delta \_{Ji}x\_{i}-X\_{J}=x\_{J}-X\_{J}}

### Displacement gradient tensor

The partial differentiation of the displacement vector with respect to the material coordinates yields the *material displacement gradient tensor* **∇XU**. Thus we have:

u

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t
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=

x

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−

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∇

X

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∇

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x

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I

∇

X

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=

F

−

I

{\displaystyle {\begin{aligned}\mathbf {u} (\mathbf {X} ,t)&=\mathbf {x} (\mathbf {X} ,t)-\mathbf {X} \\\nabla \_{\mathbf {X} }\mathbf {u} &=\nabla \_{\mathbf {X} }\mathbf {x} -\mathbf {I} \\\nabla \_{\mathbf {X} }\mathbf {u} &=\mathbf {F} -\mathbf {I} \\\end{aligned}}}

u

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=

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i

−

δ

i
J

X

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=

x

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∂

u

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∂

X

K

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∂

x

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∂

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δ

i
K

{\displaystyle {\begin{aligned}u\_{i}&=x\_{i}-\delta \_{iJ}X\_{J}=x\_{i}-X\_{i}\\{\frac {\partial u\_{i}}{\partial X\_{K}}}&={\frac {\partial x\_{i}}{\partial X\_{K}}}-\delta \_{iK}\\\end{aligned}}}

where **F** is the *deformation gradient tensor*.

Similarly, the partial differentiation of the displacement vector with respect to the spatial coordinates yields the *spatial displacement gradient tensor* **∇xU**. Thus we have,

U

(

x

,
t
)

=

x

−

X

(

x

,
t
)

∇

x

U

=

I

−

∇

x

X

∇

x

U

=

I

−

F

−
1

{\displaystyle {\begin{aligned}\mathbf {U} (\mathbf {x} ,t)&=\mathbf {x} -\mathbf {X} (\mathbf {x} ,t)\\\nabla \_{\mathbf {x} }\mathbf {U} &=\mathbf {I} -\nabla \_{\mathbf {x} }\mathbf {X} \\\nabla \_{\mathbf {x} }\mathbf {U} &=\mathbf {I} -\mathbf {F} ^{-1}\\\end{aligned}}}

U

J

=

δ

J
i

x

i

−

X

J

=

x

J

−

X

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∂

U

J

∂

x

k

=

δ

J
k

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∂

X

J

∂

x

k

{\displaystyle {\begin{aligned}U\_{J}&=\delta \_{Ji}x\_{i}-X\_{J}=x\_{J}-X\_{J}\\{\frac {\partial U\_{J}}{\partial x\_{k}}}&=\delta \_{Jk}-{\frac {\partial X\_{J}}{\partial x\_{k}}}\\\end{aligned}}}

## See also

* The deformation of long elements such as beams or studs due to bending forces is known as *deflection*.
* Euler–Bernoulli beam theory
* Deformation (engineering)
* Finite strain theory
* Infinitesimal strain theory
* Moiré pattern
* Shear modulus
* Shear stress
* Shear strength
* Stress (mechanics)
* Stress measures