Holpzaphel_-_Nonlinear-Solid-Mechanics-a-Contin

2.4 Deformation Gradient

One goal of th.is section is to study the deformation (i.e. the changes of size and shape) of a continuum body occurring when moved from the reference configuration Q0 to some current configuration n. Fo.r simplicity, we often omit the arguments of the tensor quantities in subsequent considerations.

Deformation gradient. As we know from Section 2.1, a typical point X E 0 0 identified by the position vector X maps into the point x E n with position vector x.. Now we want to know how curves and tangent vectors deform.

Consider a material (or undeformed) curve x = r(~) c no, (..YA = rA(~)), where f, denotes a parametrization (see Figure 2.4_). The material curve is associated with the reference -configuration n0 of the continuum body. Hence, the material curve is not .a function of time. During a certain motion x the material curve deforms into a spatial (or deformed) curve x = 1(~, t) c n, (:z;a = 1'n(~, t) ), at ti.me'/:.

"Figure 2.4 Defonnation of a material curve r c no into a spatial curve I C n.

The spatial curve at a fixed time t is then defined by the parametric equation

x = 1'(Ct) = x(r(€)J) or

We denote the spatial tangent vector to the spatial curve as dx and the material tangent vector to the material curve as dX. They are de-fined by

with the abbreviation (e )' = D( •) / D~. In the literature the tangent vectors dx and dX, vihich are infinitesimal vector elements in the current and reference configuration (see

4

• ':" ." , • ~ • • ~ • • , •, , ... , , : . • ( •• , : . , .. , • , , I "• "'. , "• :

3 ,., ..,.., ........... ,., ... .

1.-'----....

.··ao

bar·z .•·..: ·· . • . •••

Figure 2.5 Deformation of a unit square showing the map x of a material line element a0 into a and the inverse map x-1. of a spatial line element b into b0 •

The second-order tensor GraclU is called the d·isplacement gradient tensor in the mate.rial descr~ption. From (2.5) we find using definition (2.40) that

gradu = gradx - gTadX(x, t)

where the second-order tensor gradu is called the displacement ·gradient tensor in

the spatial description.,.

74 2 Kinematics ·

Note the following relationships between the material gradient and ·mate.rial divergence, and the spatial gradient and spatial divergence of the smooth scalar, vector and tensor fields, <I>, u .and A~ respectively. The fields are defined on the current configura-

.ample, X, r, dX, map onto points, curves, tangent vectors x, I~ .dx~ respective-ly. An arbitrary differential vector maps via the deformation gradient F (see the linear tra.ns.formation(2.3

However, .a unit vector N normal to an infinitesimal material (or undeformed) sur~ face e:Jcment dS does not map to a unit vector n normal to the associated infinitesimal spatial (or deformed) surt·ace element els via F, as shown in the following.

We perform the change in volume between the reference and the current configuration at

in which J is the detenninant of the deformation gradient F, known as the volume ratio (or Jacobian determinant). In (2.50), dl/ and dv denote infinites·imal vo]ume ele.ments defined in the reference and current configurations called material (or un.. deformed) and spatial (or deformed) volume elements, respect.ive.ly. Further, we assume that the volume is a continuous (or at least .a p·iecewise continuous) function of continuum particles so that dV' = d....Y1d..\"'2d.Y:1 and dv = fh1d:r:2d:r.:_1 (continuum idealization).

Since :F is invertible we have J(X, t) = det.F(X,.t) # 0. Because of the im-

penetrabiJity of matter, i.e. volume elements cannot have· negative volumes, we re- ject J (X., t) < 0 which mathematically is possible. Consequently, the volume ratio

J (X, t) > 0 must be greater than zero for all X E fl0 and for all times t. The inverse

of relation (2.5]) follows with identity (l.109) as J- 1 =

-

introduced in eq. (2..40)..

If there is no motion (F = land x = X), we obtain the consistency condition J = 1~ since det.F = detl = .L However, a motion (or a defonnation) with J = 1 (at every

p.artide in every configuration and time t) is called isocho,ric or volume-preserving. It keeps the volume constant.

In order to compute the relationship between the unit vectors n and N -consider an arbitrary material line element dX, which maps to dx during a certain motion X· We now ·express the infinitesimal volume element in the current con.figuration dv as a dot

with ds = dsn and dS = dSN denoting vector elements of infinitesimally sma]) areas

Wmch shows how the vector elements of the infinitesimally small areas ds and d~ on the current and .reference configurations are related. Relationship (2.54) is well-known as Nanson's formula.

EXERCISES.

..- 1. Consider a parametric curve in space of the form X = r(~), with .

de-fining a ·helix. Find the length of the helix for 0 < ~ < II Cf lr'(c;)Id~).

.: :· 2. In a deformation of a three-dimensional problem, the displac-ement components

of u are found to be

Compute the matrix repres.entations of F_,_1 and F and deduce that the defonnation is isochoric.

3. Let a -continuum body undergo a homogeneous de.fonnation whic.h is defined by where the components of the vector c and the tensor

A a.re constants or time9'.dependent functions ..

surface in the current configuration initially lies on the surface of an e] lipsoid.

(c)Consider (b) and detennine how a unit nonnal N of an infinitesimal small area dS in the reference configuration deforms. Take N = 1/v'3 (el - e2+ eJ).

2.5Strain Tensors

We have l~arnt from the ·preceding section that the deformation _gradient is the fundamental kinematic (second-order) tensor in finite deformation kinematics that characterizes changes of .material elements during motion. The aim of this section is to determine these changes in the form ·of (second-order) strain tensors related to either the reference or the current configuration.

Note that unlike displacements, which are measurable quantities, strains are based on a concept that is introduced to simplify analyses. Therefore, numerous definitions and names of strain tensors have been proposed in the literature. We discuss (and compare) the most com1non definitions of strain tensors established in nonlinear continuum

mechanics.

Material strain tensors.. We compute the change in length between two neighbor- ing points X and Y, located in region n0 , occurring during a motion (see Figure 2.6).

By neighboring we mean that X is 'close' to Y.

The geometry in the referen.ce co11figuratio11 is given by

We denote the (material) length of the material line element dX = Y - X by de. It is

d~'~::.~tw;::::::::.':):r::,•

:<~~.,"!,,!i:::r:::::::i~;:I:::

time t

', .~:."'..·. :.:. :_....·...·..:_:·.. ,.......

'.., .....:_-.:_.:_:·.:··.'..::·,-,:.:.:·

.,.... ~:\;:,'/-:·..:....... ·

Au0 = ·Fao

fAl)ol =A

time ·t = 0

laol = 1

Figure 2.6 Deformation of a material line element with Jeng.th de into a spatial line element with length Ade.

describes the direction of the material line element (which may be imagined as a fiber), as illustrated .in Fi_gure 2.6. Hence., additional.ly, we find using (2.56).1 that

Note that the vector quantities dX and n0 are naturally associated with the reference

~on.figuration of the body.

Certain motions transform the two neighboring points X and Y into their displaced positions x = x(X, t) and y = of region .Q, respectively. We now ask how close is x to y. Using Taylor's expansion according to (1.238}, {.l.23.9), y may be expressed by means of (2.55), (2.56) and the deformation gradient (2.39), as

78 2 Kinematks

where the Landau order symbol o(Y-X) refers to a small error that tends to zero faster than Y - X ~ o. With motion x = x(X, t) and (2.56)2, (2.56)3 it follows subsequently from (2.58) that

y - x = dcF(X, t).a0 + o(Y - X)

which clearly shows that the term F(Y - X) linearly approximates the relative motion

.y - x. The ·more Y approaches X the better is the approximation.$ the smaller is de- =

1v-x1.

Next, we define the stretch vector .Aa0 .in the direction of lhe unit vector a0 at

with length ,.\ = l.A110 Icalled stretch ratio or simply the stretch (see Figure 2.6). Then, the length of a spatial line element (originally in the direction of a0 ), Le. the. distance

between the two neighboring places x and y., is obtained from (2.59) . by neglecting

1

terms of order dc:2 • Using definitions (1.15) 1 and (2.60) we find with y - x:;::: dc-Fa0 that

(2.61)

.In summary: a material line element dX at .X with length de at time ·t = -0 becomes the length Ade at time "t. The stretch ,\ is a m·easure of how much the unit vector a0 has stretched. We say that a line element is extended, unstretched or com-pressed according to A > 1, ,.\ :;:::; 1 or A < L. respectively.

With definitions (l..15h, (2.·60) and property (1..8.1), the square of,.\ .is computed according to

,\2 = Ano · A110 == Fao · Fao

where we have introduced the right Cauchy-Green tensor C as an important strain measure in material coordinates (F is on the right). Frequently in the literature C is referred to as the Green deformation tensor. From (2.63) we learn that to determine the stretch of a fiber one only needs the direction a0 at a point X E 0 0 and the se-cond- order tensor C.

Note that C is symmetric and positive definite at each X E f20 . Thus,

and u · Cu > 0 for all u =!= o . (2.64)