In post 20.20, we anticipated that there must be some relationship between the Young’s modulus, *E* (post 20.2) and the bulk modulus, *B* (post 20.20) of a material. The relationship is given by

*E* = 3*B*(1 – 2*ν*) (1)

where *ν* is the Poisson’s ratio (post 20.5) of the material. We shouldn’t be surprised by the appearance of Poisson’s ratio because it relates three-dimensional changes in deformation (involved in the definition of *B*) with a stress that is applied in a single direction (involved in the definition of *E*).

Equation 1 is valid only for

- Small strains
- Materials whose properties are isotropic
- Homogeneous materials.

If the strains are small, then the material will obey Hooke’s law reasonably well, over the small range of strains considered (post 20.3). An *isotropic* material has the same properties in different directions, so *E* and *ν* will be the same in all directions. A homogeneous material has the same composition throughout – it is not a mixture of different components; so equation 1 does not apply to materials like concrete, plastics reinforced with glass fibre, bread dough and a lot of other everyday materials.

The rest of this post is concerned with the derivation of equation 1. This derivation is tedious and if you are only interested in the results, stop reading here!

__Matrix equation for relationship between stress and strain__

Don’t worry if you don’t know anything about matrix algebra – I will try to explain what you need to know in this post.

Let’s think of the deformation of a cube, whose sides have a length *L*, that defines the *x*, *y* and *y* axis directions (appendix 2 post 16.50).

What happens when we apply a tensile stress, *σ** _{x}* (post 20.2), in the

*x*-direction? A strain,

*ε*

*is generated in this direction where*

_{xx}*ε** _{xx}* =

*σ*

*/*

_{x}*E*

as explained in post 20.2. But there is also a strain, *ε** _{xy}*, generated in the

*y*-direction because of the Poisson’s ratio effect (post 20.5). By definition

ν = – *ε** _{xy}*/

*ε*

_{xx}as explained in post 20.5, so that

*ε** _{xy}* = – ν

*ε*

_{xx}= – ν

*σ*

*/*

_{x}*E*.

Similarly, there is a strain *ε** _{xz}* generated in the

*z*-directions given by

*ε** _{xz}* = – ν

*ε*

*= – ν*

_{xx}*σ*

*/*

_{x}*E*.

So the total strain induced by the stress *σ** _{x}* is given by

*σ** _{x}*/

*E*(in the

*x*-direction) – ν

*σ*

*/*

_{x}*E*(in the

*y*-direction) – ν

*σ*

*/*

_{x}*E*(in the z-direction).

Using the same arguments and the same notation, the total strain induced by a stress *σ** _{y}* in the

*y*-directions is given by

*σ** _{y}*/

*E*(in the

*y*-direction) – ν

*σ*

*/*

_{y}*E*(in the

*x*-direction) – ν

*σ*

*/*

_{y}*E*(in the z-direction).

And for a stress *σ** _{z}* in the z-direction, the total strain is

*σ** _{z}*/

*E*(in the

*z*-direction) – ν

*σ*

*/*

_{z}*E*(in the

*x*-direction) – ν

*σ*

*/*

_{z}*E*(in the y-direction).

Now let’s collect the strains in the *x*-direction together. Then

*ε** _{x}* =

*σ*

*/*

_{x}*E*– ν

*σ*

*/*

_{y}*E*– ν

*σ*

*/*

_{z}*E*= (1/E)(

*σ*

*– ν*

_{x}*σ*

*– ν*

_{y}*σ*

_{z}).

We can do the same thing for the *y* and *z*-directions so that the relationships between the stresses and strains are given by the three equations below.

*ε*_{x} = (1/E)(*σ** _{x}* – ν

*σ*

_{y}– ν

*σ*

*).*

_{z}*ε*_{y} = (1/E)(- ν*σ** _{x}* +

*σ*

*– ν*

_{y}*σ*

*).*

_{z}*ε** _{z}* = (1/E)( – ν

*σ*

*– ν*

_{x}*σ*

*+*

_{y}*σ*

*).*

_{z}Matrix notation allows us to collect these three equations together so that

In this matrix equation the strains are contained in a 3 × 1 matrix and are calculated by *pre-multiplying* the 3 × 1 stress matrix by a 3 × 3 matrix. A matrix doesn’t have a value; it consists of elements, each with their own value. You can see how pre-multiplication of a 3 × 1 matrix by a 3 × 1 matrix works by comparing this matrix equation with the three equations above it. But I provide more explanation in the appendix.

Writing the three independent equations as a single matrix equation may seem pointless. But it is often more convenient to use matrix notation rather than have to deal with several related equations. I shall also use equation 2 in other posts.

__Relationship between engineering strain and volumetric strain__

Originally our cube had a volume given by

*V* = *L*^{3}.

Now each length is increased by a strain. So, in the *x*-direction, the new length is *L*(1 + *ε** _{x}*) (post 20.2). In the

*y*and

*z*-directions the new strains are

*L*(1 +

*ε*

_{y}) and

*L*(1 +

*ε*

*), respectively. Therefore, the volume of the strained cube is*

_{z}* V*’ = *L*^{3}(1 + *ε** _{x}*) (1 +

*ε*

*)(1 +*

_{y}*ε*

*) =*

_{z}*L*

^{3}(1 +

*ε*

*)(1 +*

_{x}*ε*

*+*

_{z}*ε*

*+*

_{y}*ε*

_{y}*ε*

*)*

_{z}= *L*^{3}(1 + *ε** _{z}* +

*ε*

*+*

_{y}*ε*

_{y}*ε*

*+*

_{z}*ε*

*+*

_{x}*ε*

_{x}*ε*

*+*

_{z}*ε*

_{x}*ε*

*+*

_{y}*ε*

_{x}*ε*

_{y}*ε*

*) ≈*

_{z}*L*

^{3}(1 +

*ε*

*+*

_{x}*ε*

*+*

_{z}*ε*

*).*

_{y}The final result arises because, for small strains, one strain multiplied by another is negligibly small.

So the change in the volume of the cube is

Δ*V* = *V*’ – *V* = *L*^{3}(*ε*_{x} + *ε** _{z}* +

*ε*

*).*

_{y}And the volumetric strain (post 20.20) is

*Θ* = Δ*V*/*V* = Δ*V*/*L*^{3} = *ε*_{x} + *ε** _{z}* +

*ε*

*. (3)*

_{y.}__Matrix equation for isometric compression__

If the cube is subjected to isometric compression, the volumetric stress is given by –*p* (post 20.20). Then equation 2 becomes

Frome this equation we can see that

(1/*E*)(-*p* + *νp* + *νp*) = *ε** _{x}* =

*ε*

*=*

_{y}*ε*

_{z}=

*ε*. (5)

These equations define an isometric strain, *ε*, that is given by

*ε* = -(*p*/*E*)(1 – 2*ν*). (6)

From equations 3 and 5

*Θ* = 3*ε*. (7)

From equations 6 and 7

*Θ* = -3(*p*/*E*)(1 – 2*ν*). (8)

In post 20.19 we defined *B* by

*B* = –*p*/*Θ* so *Θ* = –*p*/*B*. (9)

And, from equations 8 and 9

–*p*/*B* = -3(*p*/*E*)(1 – 2*ν*).

Dividing both sides of this equation by *p/E* and multiplying both sides by *B* gives

*E* = 3*B*(1 – 2*ν*).

This result is identical to equation 1, so we have proved the relationship between *E* and *B*.

In the next post, we will think about the relationship between *E* and the rigidity modulus, *n* (post 20.19).

__Related posts__

20.20 Deformation of objects – isometric compression

20.19 Deformation of objects – change in shape

20.5 Poisson’s ratio

20.3 Hooke’s law

20.2 Deformation of objects

Follow-up posts

20.22 Relation between Young’s modulus and shear modulus

__Appendix__

This appendix explains some simple matrix algebra. Remember that a matrix does not have a numerical value – it is a collection of numbers called *matrix* *elements* or simply *elements*.

*Transpose of a matrix*

We will define the 3 × 1 matrix ** A** by

This is also called a *column matrix* because it consists of only 1 column that contain the elements *a*_{1}, *a*_{2} and *a*_{3}. The *transpose* of ** A** is written

*A**and is defined by*

^{T}*A** ^{T}* is called a

*row matrix*because it consists of only 1 row. That contains the same elements as

*.*

**A**We will define the 3 × 3 matrix * B* by

This matrix has 3 rows and 3 columns and so contains 9 elements. The transpose of ** B** is

*Matrix multiplication*

To calculate ** C** =

*×*

**B****multiply the first blue element in**

*A***by the first element in**

*B***, then multiply the second blue element in**

*A***by the second element in**

*B***, then multiply the third blue element in**

*A***by the third element in**

*B**; now add the results together to get the first element*

**A***c*

_{1}in the column matrix

*. Now do the same thing with the red elements of*

**C***to get*

**B***c*

_{2}and with the green elements to get

*c*

_{3}. You will find, for example, that

*c*_{3} = *b*_{13}*a*_{1} + *b*_{23}*a*_{2} + *b*_{33}*a*_{3}.

Now you can see how equation 2 gives the three expressions for the strains *ε** _{x}*,

*ε*

*and*

_{z}*ε*

*, in the three equations above it.*

_{y}If you calculate **A**^{T} × * A* you will get a single number

*a*

_{1}

^{2}+

*a*

_{2}

^{2}+

*a*

_{3}

^{2}. But you can’t calculate

**×**

*A*

**A***. And you can’t calculate*

^{T}**×**

*A***either. In both examples the order in which the matrices are multiplied is important; we can pre-multiply**

*B**by*

**A**

*A**but we can’t pre-multiply*

^{T}

*A*^{T}by

**. Similarly, we can**

*A**post*–

*multiply*

*A*^{T}by

**but we can’t post-multiply**

*A**by*

**A**

*A*^{T}. The suffixes

*pre*– and

*post*– tell us which matrix comes first in the multiplication process.

In general, we can multiply two matrices together if the number of columns in the first is equal to the number of rows in the second. Below ** D** is an

*m*×

*n*matrix and

*is an*

**E***n*×

*p*matrix.

We can the elements of * G* =

*×*

**D***from*

**E***g*_{ij} = *d*_{i}_{1}*e** _{ij}* +

*d*

_{i}_{2}

*e*

_{2}

_{j}+ …

*d*

_{in}*e*

_{nj}where * G* is a

*m*×

*p*matrix.

*Relationship to vectors and tensors*

Conventionally, we write the three-dimensional vector ** F** in the form

* F* =

**i***F*

*+*

_{x}

*j**F*

*+*

_{y}

*k**F*

*.*

_{z}where ** i**,

*and*

**j***are mutually perpendicular unit vectors, and*

**k***F*

*,*

_{x}*F*

*and*

_{y}*F*

*are the components of*

_{z}*in the*

**F***x*,

*y*and

*z*directions (post 17.3). However, it is sometimes useful to represent a vector in matrix notation as

Then, in appendix 2 of post 17.13 we could write

The dot product * v.r* is then given by

*v*^{T}

**r**=

*v*

_{x}

*r*

_{x}+

*v*

_{y}

*r*

*+*

_{y}*v*

_{z}

*r*

*that is a scalar.*

_{z}The cross product of the two vectors ** a** and

*in post 17.37 is given by the determinant*

**b**which could be represented by the column matrix

Now let’s look again at equation 2. We would get the same relationship between stresses and strains if we wrote it as

Now we are representing the stresses and strains by a 3 × 3 matrix in which are the non-diagonal elements are zero; this is an example of a *diagonal matrix*. The off-diagonal elements are zero because, the strains resulting from application of the stress *σ** _{x}* do not depend on

*σ*

*or*

_{y}*σ*

*. In general, the state of strain in an isotropic material may be more complicated so that the strain induced by*

_{z}*σ*

*may components in the*

_{x}*y*and

*z*directions that are not so simply related to

*σ*

*. In other words, the strain depends on direction but strains in the*

_{x}*x*,

*y*and

*z*direction are not independent. Then the strain in the object is represented by a tensor instead of by a vector (post 20.2). We can represent this tensor by the a 3 × 3 matrix

If you’ve managed to read this whole post, from beginning to end, congratulations!