Mae 640 Lec9

Continuum mechanics – MAE 640 Summer II – 2009

Dr. Konstantinos Sierros 263 ESB new add [email protected]

Problem 3.3 The motion of a body is described by the mapping;

where t denotes time. Determine; (a) the components of the deformation gradient tensor F (b) the components of the displacement, velocity, and acceleration vectors, (c) the position (X1, X2, X3) of the particle in undeformed configuration that occupies the position (x1, x2, x3) = (9, 6, 1) at time t = 2 s in the deformed configuration.

Change of volume and surface • We see how deformation mapping affects surface areas and volumes of a continuum. Volume Change • First we need to define volume and surface elements in the reference and deformed configurations. • Consider three non-coplanar line elements dX(1) dX(2) and dX(3) forming the edges of a parallelepiped at point P with position vector X in the reference body B so that;

Change of volume and surface • The vectors dx(i) are not necessarily parallel to or have the same length as the vectors dX(i) because of shearing and stretching of the parallelepiped.

• We denote the volume of the parallelepiped is given;

Unit vectors along dX(i)

Change of volume and surface

• The corresponding volume in the deformed configuration is given by;

J has the physical meaning of being the local ratio of current to reference volume of a material volume element.

Change of volume and surface

Surface Change • Consider an infinitesimal vector element of material surface dA in a neighborhood of the point X in the undeformed configuration as shown below;

• The areas of the parallelograms in the undeformed and deformed configurations are;

Surface Change

Change of volume and surface

• The area vectors are given by;

• The following relations can be shown;

3.4 Strain measures Cauchy–Green Deformation Tensors • We describe a general measure of deformation of a continuous medium, independent of both translation and rotation. • Consider two material particles P and Q in the neighborhood of each other separated by dX in the reference configuration as shown below;

3.4 Strain measures Cauchy–Green Deformation Tensors • In the current (deformed) configuration, the material points P and Q occupy positions ¯P and ¯Q, and they are separated by dx. • We wish to determine the change in the distance dX between the material points P and Q as the body deforms and the material points move to the new locations ¯P and ¯Q.

3.4 Strain measures Cauchy–Green Deformation Tensors • The distances between points P and Q and points ¯P and ¯Q are given below,

C is called the right Cauchy–Green deformation tensor

• By definition, C is a symmetric second-order tensor.

3.4 Strain measures Cauchy–Green Deformation Tensors • The transpose of C is denoted by B and it is called the left Cauchy–Green deformation tensor, or Finger tensor

• Recall from Eq. (2.4.15) that the directional (or tangential) derivative of a field φ(X) is given by;

Unit vector in the direction of the tangent vector at point X

3.4 Strain measures Cauchy–Green Deformation Tensors • Therefore, a parameterized curve in the deformed configuration is determined by the deformation mapping x(S) = χ(x(S)).

Vector defined in the deformed configuration •The stretch of a curve at a point in the deformed configuration is defined to be the ratio of the deformed length of the curve to its original length.

3.4 Strain measures Cauchy–Green Deformation Tensors • If we consider an infinitesimal length dS of curve in the neighborhood of the material point X. • The stretch λ of the curve is simply the length of the tangent vector F · ˆN in the deformed configuration

Holds for any arbitrary curve with dX = dS ˆN and allows the computation of the stretch in any direction at a given point.

3.4 Strain measures Green Strain Tensor • The change in the squared lengths that occurs as a body deforms from the reference to the current configuration can be expressed relative to the original length as;

Green–St. Venant (Lagrangian) strain tensor/Green strain tensor

3.4 Strain measures Green Strain Tensor •By definition, the Green strain tensor is a symmetric second-order tensor. Also, the change in the squared lengths is zero if and only if E = 0. • In rectangular Cartesian coordinate system (X1, X2, X3), the components of E are given by;

And in expanded form…

3.4 Strain measures Green Strain Tensor

Normal strains

Shear strains

And in expanded form…

3.4 Strain measures Physical Interpretation of Green Strain Components

•Consider a line element initially parallel to the X1-axis, that is, dX = dX1 ˆE1 in the undeformed body, as shown above; • We have;

3.4 Strain measures Physical Interpretation of Green Strain Components

Solving for E11

3.4 Strain measures Physical Interpretation of Green Strain Components If we use the unit extension Λ1 = λ − 1, we have;

• If the unit extension is small compared with unity, the quadratic term in the last expression can be neglected in comparison with the linear term, and the strain E11 is approximately equal to the unit extension 1. • E11 is the ratio of the change in its length to the original length.

3.4 Strain measures Cauchy and Euler Strain Tensors •The change in the squared lengths that occurs as the body deforms from the initial to the current configuration can be expressed relative to the current length • At first express dS in terms of dx;

˜B is called the Cauchy strain tensor

The tensor B is called the left Cauchy–Green tensor, or Finger tensor.

3.4 Strain measures Cauchy and Euler Strain Tensors • We can also have;

Almansi–Hamel (Eulerian) strain tensor or Euler strain tensor

3.4 Strain measures Cauchy and Euler Strain Tensors • The rectangular Cartesian components of C, ˜B, and e in indicial notation are given by;