Motion and deformation

Let us begin with the description of the motion and the deformation of a continuum. In structural mechanics, a continuum is defined by a set of continuously distributed particles, also called material points, forming a homogeneous body Ω with closed boundary∂Ω, as depicted in Fig. 2.1. Theundeformed state of such a body at time 0 is called theinitial configuration Ω in which the location of a material point is defined by the position vector X. Now, let us assume that the body experiences a deformation over time. Then, the deformed state a of the body at timetis called current configurationϕ(Ω). Based on

2.1 Kinematics

the Lagrangian approach – which follows the motion of a particle in space and time – the position of a material point in time is described by

x=ϕ(X, t) . (2.1)

Here,ϕ(X, t) defines a nonlinear bijective mapping where the material positionX rep-resents an independent variable. Further, the spatial positionxdescribes a variable de-pending on both the material positionXand the timet≥0. In other words, the mapping ϕ(X, t) relates thematerialcoordinatesX of theinitialconfiguration Ω at time 0 with thespatial coordinates of the currentconfigurationϕ(Ω) at a fixed time instantt. The difference between the position vectorsXandxis represented by the displacementu

u=ϕ(X, t)−X . (2.2)

Consequently, the map can be expressed as a function of initial position of a particle and its displacement as

ϕ(X, t) =X+u . (2.3)

Ω

ϕ(Ω) ϕ(X, t), F

u

X

x

Y, y X, x

Z, z initial configuration

(at time 0)

current configuration (at timet)

dx dV

dv

dX

dA

da N

n

∂Ω

ϕ(∂Ω)

Figure 2.1: The motion and the deformation of a bodyΩ.

In order to provide a better understanding of the following formulations and relations, we give a brief summary of theinitialas well as thecurrentconfigurations, respectively.

• Theinitialconfiguration Ω – also calledreference,undeformed, ormaterial configu-ration – refers to the undeformed and stress-free state of a body at time 0. In this

2 Basic elements of continuum mechanics

configuration, the position of the material points is defined by the initial (or mate-rial) coordinatesX which represent an independent variable. This means thatX does not change over time. In the following formulations, the usage of capital letters for quantities or operators is associated with theinitial configuration – e.g. Gradu defines the gradient of the displacement vector with respect to theinitialcoordinates (grad_Xu).

• Thecurrent configurationϕ(Ω) – also called deformed orspatial configuration – refers to the deformed state of a body at the current timet. In this configuration, the position of the material points is defined by the spatial coordinatesxwhich represent a variable depending on both the material coordinates X and the time t. The dependency is described by the nonlinear bijective mapping functionx=ϕ(X, t).

In the following formulations, the usage of small letters for quantities or operators is associated with thecurrent configuration– e.g. gradudefines the gradient of the displacement vector with respect to thecurrentcoordinates (grad_xu).

In order to study the deformation (the change in size and shape) of a body when it is mapped from the initial to the current configuration, let us introduce thedeformation gradientF – which, in continuum mechanics, is an important quantity when it comes to describing local deformation processes. To this end, we consider an infinitesimal vector dX in the initial configuration, connecting two material pointsX andX+ dX. Thus, the vector dXcan be interpreted as an infinitesimal line segment at pointX. The related deformed vector dxin the current configuration is defined by the current positions of these material pointsϕ(X, t) andϕ(X+ dX, t), respectively. The transformation of the initial vector dXto the current vector dxis defined by the gradient of the deformation map

dx= ∂x

∂XdX= Gradϕ(X, t) dX . (2.4)

Thus, with the definition of the deformation gradientF

F= Gradϕ(X, t) (2.5)

Eq. (2.4), finally, reads

dx=FdX . (2.6)

Consequently, the deformation gradientF represents a linear operator that maps an in-finitesimal vector dX from the initial configuration to its counterpart dxin the current configuration. Further, to preserve the connection of the body during the deformation pro-cess and to avoid self-penetration, we postulate the following condition for the determinant of the deformation gradient

J= detF≥0 . (2.7)

Moreover, the deformation gradient can be also formulated in terms of the displacements by utilizing the relation stated in Eq. (2.3). Then, the deformation gradientF reads

F= Gradϕ(X, t) = GradX+ Gradu=1+H , (2.8) whereHdefines thedisplacement gradientand1denotes thesecond-order identitytensor.

Now, having introduced the deformation gradient allows us to describe the transforma-tion of further geometric quantities such as surface and volume elements. To this end, let

2.1 Kinematics

us consider an area element dAwhich is located at the surface∂Ω of the undeformed body Ω. Further, let us assume that the area dAis defined by two infinitesimal and linearly independent vectors dXand dY describing tangents on an arbitrary point at surface∂Ω.

Note that vector dXis not the same as in Fig. 2.1. Thus, the area element in the initial configuration can be described by the cross product of the tangent vectors as

NdA= dX×dY , (2.9)

whereN is the unit normal to the tangents and dAdefines the area. Now, using the transformation relation in Eq. (2.6), we can map the tangent vectors form the undeformed surface∂Ω onto the deformed oneϕ(∂Ω). Thus, the deformed area element in the current configuration can be described as

nda= dx×dy=FdX×FdY , (2.10)

wherenrepresents the unit normal to the deformed tangents and dadefines the deformed infinitesimal area. Finally, the area elements of the initial and the current configuration are related to each other by the well-known Nanson formula

nda=JF^−TNdA . (2.11)

Next, let us discuss the change in volume of a body. To this end, we consider an infinitesimal volume element dV at an arbitrary point within the inside of the undeformed body in the initial configuration. In doing so, the volume element is defined by three infinitesimal and linearly independent vectors dX, dY, and dZ. Observe that dX does not represent the vector given in Fig. 2.1. Then, the volume of the undeformed element can be described as

dV = (dX×dY)·dZ . (2.12)

Using the relation in Eq. (2.6), the deformed volume element dvin the current configuration reads

dv= (dx×dy)·dz= (FdX×FdY)·FdZ . (2.13) Thus, from Eq. (2.12) and (2.13), we can deduce the following mapping that relates the undeformed volume element dV with the deformed one dv

dv= detFdV =JdV . (2.14)

Having described the local deformation process by introducing the deformation gradient F, let us conclude this section by mentioning some important transformation rules that are needed to performpush forward andpull back operations. From a theoretical point of view, there is no difference to describing the basic relations of continuum mechanics with respect to the initial or the current configuration. Thus, formulations in the current configuration can be transformed to the initial one by applyingpull backoperations and vice versa by applyingpush forward operations. Some important transformation rules of thepush forwardoperations concerning the gradient and divergence operators are

gradβ=F^−TGradβ , gradβ= Gradβ F , divβ= 1

JDivβ , (2.15)

2 Basic elements of continuum mechanics

whereβrepresents a scalar field andβis a vector field. Further rules characterizing the integral transformations are given as

Z Finally, the corresponding transformations of thepull back operations are obtained by reformulation of the above expressions.