Review of Basic Elasticity Equations

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Review of Basic Elasticity Equations

Stresses and Cauchy's Stress Formula

6 Stresses: ¬ 9 terms, but for most analyses

plane direction

Slice cube to obtain tetrahedron

Boundary Tractions: (Cauchy’s Stress Formula)

Components of outward normal

Details of derivation of Cauchy’s formula

The derivation is based on equilibrium in directions.

Cauchy’s formula assures us that the nine components of stresses are necessary and sufficient to define the traction across any surface element in a body. Hence the stress state in a body is characterized completely by the set of quantities. Since is a vector and Equation (3.4-2) is valid for an arbitrary vector , it follows that is a tensor. Henceforth will be called a stress tensor. p. 72 Y.C. Fung

Details

plane direction of traction

Define = area of face of tetrahedron which has outward normal in direction.

Forces on 3 of the 4 forces of tetrahedron

Net force = net force in direction j

We can prove that

Þ sum of force in 3 faces

Force on 4th face

Sum of forces = 0

Þ

FACE AREA

Unit Normal to

Þ Note that

Example: Bar Under Uniaxial Stress

· 

· 

What is the traction on the plane AB? (see sketch below)

Matrix form

·  What is net force in x1 direction?

If bar had been subjected to pure shear , then tractions on the plane AB would be

Derivation of Equilibrium Equations:

Strains and Compatibility

Tensor Strains (linear):

Engineering Strains: Same as tensor except 2 ´ for shear strains

(we will usually use engineering shear)

Linear Shear Strain

Tensor shear strain = 1/2 x change in angle between two initially perpendicular lines

Engineering shear strain = change in angle between two initially perpendicular lines

2D Example

Compatibility Equations

·  Why needed?

3 displacement functions

6 strain functions

But strains are derivable from displacement

Þ cannot have 6 independent strain functions

Þ need 3 constraint equations.

[There are 6 compatibility equations but only 3 independent ® can choose any 3]

These are one possible choice.

·  Not needed if formulated in terms of displacements

·  We will probably not need

Engineering Properties

Define the following by describing a suitable experiment

·  Young’s modulus (extensional modulus)

·  Shear modulus

·  Poisson’s ratio

·  Thermal expansion coefficients

Note: The meaning is not clear for other than isotropic materials and orthotropic materials in the material coordinate system. … more later

There are many other properties, but these will be our primary focus.

General Linear Constitutive Relations

·  Relate stress and strain tensors

·  In the absence of thermal and moisture effects,

Symmetry of strain tensor:

(see definition of strain)

Symmetry of stress tensor:

If there exists a strain-energy density function U such that, then,

Requirement for positive definiteness (wait until we get to 2D analysis)

Contracted Notation (Voight)

only 6 unique stresses

only 6 unique strains

Let’s just use 1 index and have range

Similarly, for use 2 indices with range

6 unique possibilities for each pair of indices

Comments

·  Don’t forget the components are parts of tensors. The lists of stresses and strains and the array of the constitutive coefficients are not tensors.

·  When we use engineering shears strain, the components are not part of a tensor but transformations can be derived from the relationship between engineering and tensor shear strains.

Contracted Notation & Engineering Shear Strain (order can vary!)

If x, y, z ® 1, 2, 3

Note:

General:

Content of C (ie. non-zero Cij) depends on material symmetries

Boundary conditions

·  Specify traction or displacement along entire boundary

·  Cannot specify at same point, etc.

Note: You do not specify strains or components of the stress tensor.

Self-study

Index Notation

1.  Repeated index indicates summation

2.  Index appearing more than twice has no meaning. Do not do it.

3.  Range of summation should be noted if it is not clear from the context.

4.  Summation indices are called “dummy indices,” i.e. they can be replaced with another index.

5.  Unrepeated indices are called “free indices”. The free indices must be the same in every term.

6.  Kronecker Delta:

Examples

Equilibrium: