ABSTRACT
This chapter is devoted to the basic description of strains and stresses existing in solids and addresses principal experimental methods and techniques used to measure them� This may be regarded as the introduction into the resistance of materials, which is one of the most difficult engineering disciplines to understand and master� There are many classic textbooks on this subject, some of which we are citing and referring to [1-24], alongside with numerous priority publications relevant to physical-chemical mechanics [25-41]�
The material is presented in such a way that it is oriented toward the audience that does not have extensive specialized knowledge in the area of mechanics of a rigid body� Yet, knowledge and understanding of the principles discussed in this chapter is essential in making a transition from the discussion of the physical-chemical mechanics of coagulation structures to the discussion of physical-chemical mechanics of solid-like systems�
We will start the discussion by introducing the concepts of force and stress, which will lead to a discussion on the comparison of a vector to a tensor� In physics, the term “force” automatically implies that there are means to measure it�
Let a force F be applied to a body� Let us draw an origin through the point at which the force is applied and then replace the force with two forces, Fx and Fy, measured using two calibrated dynamometers along the two chosen axes� This scheme allows us to introduce the concept of the components of a force�
Now, let us turn the coordinate system by an arbitrary angle, θ� The projections of the force components on to the new axes, x′ and y′, can be calculated from the original force components and direction cosines as
F F F
F F F
= ¢ + ¢
= ¢ + ¢
cos(x x) cos(x y) cos(y x) cos(y y)
The transform matrix contains the following components:
cos( ) cos( ) cos( ) sin cos(
¢ ¢ ¢ ¢
= = = = = =
x x
y y x cos y sin x
q q q
a a
a
21 y cos ) = =q a22
where indices 1 and 2 correspond to x and y, respectively� The first index is related to the new axes and the second one to the old axes� Using the components of the transformation matrix, the transformations of the force components can be written as
F F a F a F a F
F F a F a F a
= ¢ = + =
= ¢ = + =
Fj
The above expressions can be generalized as ¢ = =åF a F a Fi ij j ij jj , where the repeating index automatically implies summation�
The essence of this transformation is that the components of force transform linearly with the turn of the axes, that is, they can be represented by the sums of the initial components, with the coefficients given by the direction cosines to the first power� This establishes the characteristic of the force as a physical variable� We have also defined a vector as a first-order tensor� The first order of the tensor is defined in this case by the first power of the cosines in the transformation equation� A vector can be multidimensional while still remaining a first-order tensor� A scalar can be defined as a tensor of zeroth order�
Force is a field vector, characterizing the presence of an external force field� Field vectors do not represent specific characteristics of materials, but they describe interactions in the materials, the motion of materials, etc� There are also material vectors, one example of which is the pyroelectric vector showing polarization in the body upon heating�
Let us turn again to the decomposition of a force into components (Figure 5�1)� Does one need to have exactly two components (in 3D space, three components), or can this representation be simplified? In general, one does indeed need two components� However, one of them may disappear when the axes are turned in a certain way� Indeed, one of the components becomes equal to zero when the origin is turned by an angle θ such that
F F F¢ = + =y x y( sin ) cos-q q 0
tan or arctany
q q= = æ
è ç
ö
ø ÷
F F
F F
In such (and only in such) a coordinate system, one will have F F F F¢ = + =x x y2( ) ./2 1 2 The coordinate system in which this is true is said to be formed with principal axes� In these coordinates, the length
of a vector, that is, its modulus F F= ( )å i2 1 2/
, is the invariant (Inv) of a vector in all axes�
In order to fully understand the meaning of a vector, it is essential that we examine all these transformations in detail in various coordinate systems� A force can be easily identified using a picture of a load hanging on a rope� The rope shows the force direction, while the load indicates the force magnitude�
The situation is more difficult if one is measuring the strength of a magnetic field after removing a piece of wire from a coil: in this case, one can only observe the direction in which the galvanometer arrow moves rather than the direction of the field� The situation is even more difficult in the case of multidimensional vectors that one can’t represent with a drawing in a 3D space� Nevertheless, the concepts of a vector variable and a nonvector variable are quite rigorous: if the components reflecting the action imposed on a body or the properties of a body follow the described transformation scheme, then the property is a vector quantity and one can make a transition to the principal coordinate system in which the “arrow” will symbolize the magnitude and direction of a given physical quantity�
Let us go back to the case where a body undergoes deformation or strain as a result of an applied force� All interatomic bonds get deformed, and the atoms are displaced from their equilibrium positions� If originally the body had a rectangular shape, after the deformation, its shape has changed, that is, the body was stretched and sheared� As a result of the strain, a field of mechanical stresses has been generated� This field has produced localized stresses that penetrate the entire body� These stresses are compensated overall by the applied force, F� How can one describe these internal forces? To do this, let us employ the same approach as was used in describing the components of a force vector� Within a given coordinate system associated with the body, one can select elementary flat platforms that have a unit area� One now needs to find a way to estimate the forces that act on these areas� Let us use the symbols x and y to denote platforms perpendicular to the x and y axes, respectively� Let us make incisions along these platforms (Figure 5�2)� In order to maintain the material on both sides of the incisions in place, one needs to apply an additional force to compensate for the broken (“cut”) bonds� The forces that need to be applied can be measured with spring dynamometers� Let us restrict ourselves to a 2D model� Let us mount a dynamometer in a cut x in the direction x� This dynamometer will measure the stress component σxx� In this notation, the first index refers to the platform and the second one indicates the force direction� It is rather remarkable that in colloid chemistry, σ is commonly used for surface tension, while in mechanics, it is used to denote stress, and in physical-chemical mechanics, it is used for both! Similarly, the dynamometer in a scission y applied in the direction y will measure the stress σyy�
However, it is not sufficient to use just the two springs that we have so far utilized� One needs to also compensate for the tangential (shear) forces� To do this, two more spring dynamometers parallel to the plane of scission are needed� These dynamometers will measure the stress components σxy and σyx� These components are measured in the plane x in the direction y and in plane y in the direction x, respectively� We have thus obtained a combination of four variables that completely describe the stressed state in the selected 2D model�
Our model will be more informative if we completely separate out a unit cube and apply to the unit –x and –y planes (in the “left” and “lower” scissions) an equilibrating system of four forces that are pairwise symmetric with respect to components in x and y planes� This will allow us to exclude both cube translation and rotational motion in the plane of the figure (around the z-axis, which we have not yet shown)� The latter means that the shear components are identical, that is, that σxy = σyx and their net momentum around the z-axis is zero� The matrix containing the components of the stressed state is thus symmetric: only three components in it are independent:
s s s s
æ
è ç
ö
ø ÷
At the same time, we have concluded that we need four spring dynamometers� The dynamometers measuring the tangential forces are equally stretched (or compressed), but these forces belong to different platforms (as required by Newton’s third law)� This does not look like a vector at all�
When the components of the stressed state are the same, that is, completely coincide with each other, one can talk about a uniform stressed state� If any one of these components is different from the others, one has a nonuniform stressed state� In this section, we will address only the uniform stressed state�
Now, let us make a transformation to a different coordinate system by turning the axes by an angle θ, similar to how we did for a vector� The peculiarities of this transformation reveal the specifics of the stressed state as the second-order tensor�
Let us use the following two-step example (Figure 5�3)� Let us cut our cube parallel to the new axis y′, that is, the cut is in the y′Oz plane� Such a cut separates a prism out of the cube� In the plane of the figure, this prism is represented by a right-angled triangle� Let us choose the section in such a way that the length of the hypotenuse and consequently the area of a section in the perpendicular plane are equal to one� Let us mount two springs in this cut-one normally and the other one tangentially-that would compensate the components s¢xx and s¢xy� In Figure 5�3, the two components that are depicted by arrows are not applied to the hypotenuse (left and bottom) but to the opposite side of the section (right and top)� This is done in order to provide a more clear illustration of the summation procedure that we are about to consider�
We know the four forces (stress components) acting on the catheti: by projecting them on the hypotenuse, one finds both the s¢xxand the s¢xy components in the new coordinates� First, one needs to recognize that not the entire force is applied to the right cathetus but only part of that force, which is proportional to the cathetus length and the cosine of the angle θ between the x and x′ axes, that is, σxx cos θ� Consequently, the normal force applied to the upper cathetus equals σyy sin θ� The same is true for the tangential components� The transformation matrix for the projection of all four “partial” components on the new x′ axis can be written as
cos x cos y cos x cos y
cos sin cos sin
( ) ( ) ( ) ( )
¢ ¢ ¢ ¢
æ
è ç
ö
ø ÷ =
æ
è ç
x x
x x
q q q q
ö
ø ÷ +
æ
è ç
ö
ø ÷
a a
a a
In this transformation, the first index corresponds to the new axis and the second one to the old axis� From this, one gets
s s
s s s¢ ¢ = + +¢ ¢ ¢ ¢ ¢x x x x x x ¢xx
xx xy yxcos x cos x cos x cos y cos( ) ( ) ( ) ( ) (x x x xy cos x cos y cos yyy) ( ) ( ) ( )¢ ¢ ¢+ s
which is equivalent to
1 1= + + + = -
åa a a a a a a a a a k l
skl
Generalizing this approach for all four components in the new coordinate system and omitting the summation sign for repeating indices, one gets
s s s
1 2 a a
a a (5�1)
So, what does the “stress state” concept really mean? We have applied a single force (“arrow”) to the body� This force is a vector that undergoes linear transformation when the coordinate system is rotated� Linear transformation implies that the old components are summed with the transformation coefficients given by the direction cosines to the first power� Within a volume of a solid body, we have encountered a stressed state that can’t be described using a single vector�We had to isolate a “cube with scissions” and introduce a system of arrows to depict the acting forces� Each arrow represents a component that is a vector with a particular direction and absolute value� At the same time, a system of arrows is a completely different physical representation� The peculiarities of this representation are associated with how the “system of arrows” undergoes linear transformation� We have rigorously carried out these transformations and have shown that a new physical variable that we have referred to as the stressed state has several components that undergo linear transformation when the coordinate system is rotated� In the new coordinates, these components are represented by the linear combination of the old components with the corresponding transformation coefficients� These coefficients are the cosines between the new and old axes raised to the second power� This power constitutes the principle difference between stresses and vectors� Equation 5�1 represents a definition of a second-order tensor� Within the given planar example, this tensor has two dimensions, and in 3D space, it has three dimensions� In the 2D form, this tensor consists of four components with only three components being independent� In the 3D form, the tensor has nine components� These tensors are symmetric with respect to their diagonals�
Is it possible to further simplify the expression of the stressed state, similar to how it was done for a vector? Is there a coordinate system in which it can be shown with only one arrow? The former is possible, while the latter is not! Simplification in this case implies a transformation to the coordinate
system in which the shear components (tangential to the coordinate planes) will be zero� This means that all components with mixed indices are zero, that is, s s¢ ¢ij ij= = 0. In agreement with the summation rules, one gets for s¢12
s s s s s s q¢ ¢12 21= = = + + + = -0 11 21 11 11 22 12 12 21 21 12 22 22a a a a a a a a cos ( sin sin
q s q qs q q s q qs
) cos cos sin ( ) sin cos
+ + - +
This indicates that the condition s s¢12 21= =¢ 0 is required in order to preserve only the normal components in the equation of the stressed state� This condition can be met only if the coordinate system is turned by an angle θ:
q s s - s
s s - s
= é
ë ê
ù
û ú =
é
ë ê
ù
û ú½ ½
½ ½
11 22( ) ( )
As one can see, out of four components only two remain present:
s s
s s
¢ ¢
0 0
0 0
æ
è ç
ö
ø ÷ =
æ
è ç
ö
ø ÷
The right-hand notation is the conventional shortened form of the principal components of a tensor in its principal coordinate system�
It is worth recalling here that each tensor has an order (I, II, III, IV, etc�)� Tensor order reflects the physical properties of a tensor and is determined by the power of the direction cosines product, that is, the power of the product of linear transformation coefficients� The tensor order physically reflects the possibility of visualizing the various properties of a field or a body from different viewpoints� Tensor order is also an indicator of the different ways in which spatial anisotropy is revealed� Scalar quantities, that is, temperature, mass, and amount of heat, are zeroth-order tensors; the vectors of velocity or force are the first-order tensors; mechanical stresses and strains are second-order tensors, while the elasticity modulus is a fourth-order tensor, as will be shown in the following text�
At the same time, when a physical property is represented by a tensor of a given order, it can be characterized by a particular number of components in a given space� For example, the three spatial components and time form a 4D field vector, which is a first-order tensor� Another noteworthy example is that of the fourth-order elasticity tensor, which in an isotropic medium is degenerated into two scalar quantities: the Young’s modulus and the Poisson ratio�
Any physical quantity along with its spatial and geometric dimensions can also be characterized by the physical dimensionality of the components, which is the same for all components� The components of a second-order stress tensor have the dimensions of N/m2= Pa = 10 dyn/cm2 = 10−5 kg-force/cm2� At the same time, a second-order strain tensor’s components are dimensionless (of zeroth-order physical dimensionality)� This is true for both 2D and 3D models�
One can thus state a critical difference between a tensor and a vector� The aforementioned linear transformation, when applied to a vector in the principal coordinates leaves only one component� In the case of a tensor, two components are left when the transformation is applied� These components can be expressed numerically or shown graphically with arrows, but they can’t be combined, because they are applied to different platforms (planes)� Without considering the 3D case here, let us give several particular examples of stressed states in a 2D model (Figure 5�4)�
a. While a uniaxial extension represents the simplest case of a stressed state, it requires the most careful and detailed consideration� The term applies to the extension of a rod, a plate, or a cylinder but does not apply to the extension of a hook or a chain� In these latter cases, the extensions are nonuniform�
Let the object for uniaxial extension be a cylindrical rod with a cross-sectional area S to which an extension force F is applied� The ratio F/S = P is the extensional stress� Alternatively, this is a component of a stress tensor in the selected coordinate system, σyy = σ22 = σ2 = P. The action of the force resulted in a stressed state that is uniform in all directions (we are neglecting nonuniformities and local concentrations of stresses in certain special areas, such as the mounting points of the load or of the rod itself)� The selected coordinate system in this case is the principal one� From symmetry considerations at P = σyy, this tensor is as follows:
0 0 0 P
æ
è ç
ö
ø ÷
It appears that we have ended up with only a single component (“arrow”)� But in fact, this is not the case� To illustrate this, let us rotate the axes by an angle θ (e�g�, counterclockwise; a turn can also be characterized by either the “+” or the “–”sign)� The summation yielding the new components takes place over the single present component, σyy = P, which yields a new form of the same tensor:
P P P P
sin sin cos sin cos cos
q q q q q q
æ
è ç
ö
ø ÷
For a vector F, there are only two components, F cos θ and F sin θ� In Chapter 7, where we will discuss the Rehbinder effect in detail, we will be looking only
at the influence of the active media on the mechanical properties of single metallic crystals grown out of metal wires� In this discussion, we will use notations that are common for such systems� Within a given crystallographic plane, a single elliptical platform oriented by an angle χ relative to the longitudinal axis is selected� Relative to the selected directions of axes in the preceding text, χ = 90° – θ� This chosen crystallographic plane is typically a sliding or cleavage plane specific to a given crystal� With respect to this plane, there are two stress components: the extension (typically positive, “+”) component, p, perpendicular to the plane and the maximal tangential shear component, τ, applied in the direction of the larger axis of the ellipse, that is,
p P P
= = = =
s c t s c c
¢ ¢ 22
sin sin cos .
