1_1Kinematics

EN222: Mechanics of Solids

Division of Engineering Brown University

1. Basic Principles

In this section we review some basic ideas in describing deformation and internal forces in solids.

1.1 Kinematics

We describe the deformation and motion of a solid by a mapping .

Suppose a material is at position X in the undeformed solid, and moves to a position x when the solid is loaded. A mapping

would describe the motion.

We assume that χ is twice jointly differentiable wrt time and position, and satisfies

The displacement of a material point is

Homogeneous deformations

Homogeneous deformations are of particular interest to us, because the constitutive response of a solid is usually determined by measuring the forces required to induce a homogeneous deformation within a solid specimen.

A homogeneous deformation has the form

Where A is a constant two tensor such that det(A)>0, and c is a constant vector.

x

X

u

χ

Reference config Deformed config

Imagine drawing two straight, parallel lines on the reference configuration of a solid. If the deformation of the solid is homogeneous, the two lines remain straight in the deformed configuration, and the lines remain parallel. Furthermore, the lines stretch by the same amount, i.e.

Every (smooth) deformation is locally homogeneous. To visualize this statement, imagine drawing a straight line on the reference configuration of a solid. The line would look like a smooth curve on the deformed

configuration. However, suppose we focus attention on a line segment much shorter than the radius of curvature of this curve. Our segment would be straight in the reference configuration, and would also be (almost) straight in the deformed configuration. Thus, no matter how complex a deformation we impose on a solid, infinitesimal line segments are merely stretched and rotated by a deformation.

Deformation measures

To specify the constitutive response of an elastic solid, we need a measure of the deformation in the immediate neighborhood of a point in the solid. For example, we could track the deformation of a segment d X at point X in the reference configuration:

where F is the deformation gradient at X.

Actually, it is better to find a measure of the change in length of d X rather than to find its image in the reference configuration. For this purpose, note that

provides a measure of the length change.

We therefore often use the Lagrangean strain tensor (also called Green’s strain tensor) E to characterize the deformation near a point.

Some other deformation measures are

The right Cauchy-Green deformation tensor, defined as The Left Cauchy-Green deformation tensor, defined as

The Eulerian strain tensor, defined as

The right stretch tensor, defined either through the polar decomposition of

where R is a

The left stretch tensor, defined either through or computed as

The rotation tensor, which may be computed through

Principal stretches, principal strains

Consider a general homogeneous deformation with deformation gradient . Observe that since U is a

symmetric positive definite tensor it may be expressed in the form

where are three mutually perpendicular unit vectors and are called the principal stretches of the

deformation. The principal stretches are determined in practice by taking the square root of the eigenvalues of . The unit vectors are the normalized eigenvectors of .

Recall that the physical significance of the deformation gradient is that it relates the length and orientation of a material fiber in the reference configuration to its length and orientation in the deformed configuration through . The operation can therefore be visualized as a sequence of two deformations. The second (R) consists of a rigid rotation. The decomposition of U into principal values shows that the first deformation can be visualized as taking a unit cube of material from the reference solid with faces normal to

, and then stretching the cube to form a rectangle with dimensions as shown in the figure.

Any deformation therefore consists of a sequence of this stretch followed by a rigid rotation.

It is easy to see that C will decompose in the same way, with the same principal directions and principal values

.

The Lagrange strain tensor can also be decomposed into principal values and directions. The principal values

must be

The relationship between these values and principal values and directions of B, V and is left as an exercise.

The principal stretches form a convenient measure of deformation for finite strain elasticity.

Strain invariants

When defining constitutive relations for isotropic solids (which must be independent of the basis used to define the stress or strain components) it is useful to use deformation measures that are independent of the choice of basis.

Of course, the principal stretches provide one such choice. However, they cannot be expressed in terms of strain m 1

m 3

m 2

λ

3

λ1

λ

2

m

1

m 2

m

3

Instead, the most commonly used strain invariants are

which characterizes volume changes.

Usually a measure of deformation that is independent of volume changes is then introduced, as

The remaining deformation measures are then computed in terms of . In finite elasticity theories the most commonly used deformation measure is the Left Cauchy-Green strain tensor, defined as

and the two remaining invariants are then defined as

Another commonly used second invariant is

Invariants of the infinitesimal strain tensor are used in defining so-called `hypoelastic’ constitutive relations, wherein the material responds elastically (i.e. one can define a strain energy potential), but the relationship between strain and stress is nonlinear. In this case commonly used strain invariants are

Of course, other definitions can be obtained by combining these.

Measures of deformation rate

In plasticity and viscoelasticity theories we often require measures of the rate of deformation of a solid. One measure is the velocity field

It is often more convenient to express the velocity field in terms of the deformed position of material particles (that is to say, find a formula for the velocity of the material particle which is at position x at time t), which of course follows as

As a measure of deformation rate, we note that the relative velocity of two material particles with current position and can be computed from the spatial gradient of the velocity field as

leading to the definition

showing that .

It is often convenient to decompose L into symmetric and skew-symmetric parts as

Here D is called the rate of deformation tensor or the stretch rate tensor, while W is called the spin tensor.

A physical interpretation for L,D and W follows from an argument similar to the one leading to the definition of Lagrange strain tensor. Note that a measure of the rate of change of length of the material fiber d x can be obtained as

so L provides a measure of stretch rate. But note also that

where we have noted that because W is skew-symmetric ( ). Thus W is a measure of

deformation rate that causes motion without stretch – i.e. a spin. D provides a measure of rate of stretching, which is preferable to L because D vanishes for a rigid motion, while L for rigid rotations.

We can of course define other measures of deformation rate. For example the rate of change of Lagrange strain tensor

which can be related to L by noting that so that

Infinitesimal Deformations

In many problems (e.g. any linear elasticity problem and some plasticity problems) we can assume that the solid’s shape remains close to its reference configuration at all times. Specifically, we assume

with

Note that and so

We use the latter, linear measure of deformation in linear elasticity, and define the infinitesimal strain tensor

In component form

where the comma denotes differentiation with respect to a spatial coordinate, in the usual manner.

Deformation rate measures are approximated as

Some remarks on infinitesimal deformation measures

Recall that in general finite deformation kinematics, a homogeneous deformation may be decomposed into a sequence of a rotation followed by a stretch, or alternatively into a stretch followed by a rotation (the polar decomposition theorem), i.e.

Where R is a proper orthogonal tensor (representing a rigid body rotation), and U,V are symmetric stretch tensors.

There is an analogous decomposition for infinitesimal motions. Note that plays the role of F. We may write

Here w is a skew tensor, which represents an infinitesimal rotation.

To see that w represents an infinitesimal rotation, consider a deformation such that . Evidently, this requires

that

In other words, the displacement gradient is skew. Let ω be the dual vector of w, so that

which we recognize as an infinitesimal rotation.

Observe also that the decomposition of is additive. This is unlike finite deformation kinematics, where a

sequence of two deformations results in a multiplicative decomposition of the deformation gradient.