THEORY OF STRAIN In the general theory of strain, Iet P, Q be two neighbouring points of an elastic body, and P', Q', the new positions of these points when the body is deformed. Using a system of rectangular axes to specify the positions of these points, and denoting their co-ordinates by (x, y, z), (x+dx, y+dy, z+dz), (x', y', z'), (x'+dx', y'+dy', z'+dz') respectively, the differences are regarded as the component displacements of the points re spectively.
If now ds represents the small distance PQ, and ds' the small distance P'Q', the ratio of ds' —ds to ds is defined to be the strain of the element ds. In the mathematical theory, a limit ing process is supposed to be carried out in which the length PQ is made indefinitely small, and the strain is defined in very much the same way as a differential coefficient. The strain is then associated with the point P, but depends upon the direction of the element PQ. Indeed, if the space occupied by the body is assumed to be Euclidean before and after deformation, the relations in which the coefficients a, b, c, 2f, 2g, 2h depend on the partial derivatives of the displacements u, v, w. These coefficients are called the six components of strain, and may be rewritten as si, 56, or as according to cir cumstances, one notation being sometimes more convenient than another.
The six components of strain are connected by certain rela tions which imply that the expression for is suitable for the representation of the square of the element of length in Euclidean space (x, y, z, are regarded now as generalized co ordinates). When the squares and products of the partial deriva tives are so small in comparison with the derivatives themselves that they may be neglected, the expressions for the components of strain may be written in the form :— and the relations are easily found. These are called the conditions of compatibility.
The strain represented by a single component of type a, the other components being zero, is a simple extension parallel to the axis of x. The strain represented by the single component 2f, the other components being zero, is called a shearing strain. When f is constant this shearing strain may be regarded as made up of two simple shears for which the corresponding displace ments are, respectively, u=o, v=o, v=fz, w=o.
The displacement in a simple shear may be regarded as coma posed of two parts; viz., a rotational displacement, u=o, v= —kz, (2k=f), for which there is no corresponding strain, and the displacement c =o, v = kz, w=ky, for which the three components of rotation defined by the equa tions are all zero. This resolution of a displacement into a rotation and an irrotational part has an analogue in the case when the strain is not constant, for we may write du= (adx+hdy+gdz) + ('qdz —rdy) dv= (hdx + bdy+ fdz) + (-dx — dy) dw = (gdx+fdy+cdz) + (dy Invariants.—When the axes of co-ordinates are changed, the six components of strain are changed also, but certain combina tions of them remain unaltered in form. These invariants may be obtained by remarking that the ratio ds' :ds for any direction through P is inversely proportional to the central radius vector, in that direction, of the quadric surface whose equation in rec tangular co-ordinates is - = I; the lengths of the principal axes of this quadric are consequently invariants. The principal axes of the quadric are called the prin cipal axes of strain for the point P, and the extensions of a linear element for these directions are called the principal extensions E1, E2, E3. The values of i + I -I-- E2, I + E3 are the positive square roots of the three values of k for which i+2a—k 2h 2g 211 I+2b—k 2f = o.
2g 2 f 1—k i The coefficients of the different powers of in this equation2 are naturally invariants. These invariants are respectively a+b+c I2= An invariant expressible in terms of these is obtained by putting k = o in the determinant. This invariant may be written, in an alternative form, as the square of a Jacobian where A is a quantity called the cubical dilatation. When squares and products of the derivatives of u, v and w are neglected, the formula for the dilatation becomes simply The Strain Energy Function.--In 1837 George Green ob tained a generalization of Hooke's law, suitable for the most general type of crystalline substance, by starting from the as sumption that when the components of strain in a deformed body are defined by considering displacements from an initial state in which the body is free from strain, and consequently also free from stress, and when the components are also small, there is a strain energy function W which can be supposed to repre sent the density of strain energy at each point (x, y, z) of the body. In 1855 Lord Kelvin gave a thermodynamical argument which made Green's hypothesis seem plausible in two cases, first when the deformation takes place at a constant temperature and secondly when it takes place without loss of heat. Green as sumed further that there is in general a relation of type where the coefficients cmn are elastic constants characteristic of the material. For convenience it will be assumed that cnm = cmn; this may be done without loss of generality because the sum of these two quantities occurs as the coefficient of in the com plete expression for IV.
Lord Kelvin's thermodynamical argument indicates that it is necessary to distinguish between two sets of elastic constants, the isothermal and the adiabatic; it also shows that the quantities specify the state of stress in the body. The relation 2W = S1S1 + S2S2 + S3S3 + S4S4+S5S5 +S6S6 is, in fact, a natural extension of the formula 2W =SS already obtained for the case in which there is only one component of stress which is constant throughout the body and of the nature of a simple tension.
The quantities Si, S2, S3, S4, S5, S6 may be regarded provisionally as the six components of stress. They are usually denoted by another set of symbols such as , and this notation is completed by introducing quantities de fined by the relations Z,, = =4,,, Y. = The general expression for W is suitable for a crystalline body or a substance like wood, whose physical properties are related to certain definite directions in the material. For an isotropic substance, i.e., a sub stance in which all directions are alike as far as structure is concerned, the expression for IV should have the same form in all systems of rectangular co-ordinates, and so should depend only on the invariants I3. The natural assumption is thus 2W _ (A+ - 4A/2, where X and are elastic constants. The corresponding relations between stress and strain are now = 2µb, = 2µg, = 2µ11; where, as before, The relations between stress and strain may be solved for the components of strain giving, in particular, Eb= The quantity J represents the ratio of the volume of a small portion of the body after and before deformation. When the approximate expressions for the strains are adopted, it is clear that there is no change of volume in the shearing strain for which f is the only component different from zero.
In these equations E is Young's modulus, a is Poisson's ratio, is is the modulus of rigidity, and the quantity k = A+3µ is the modulus of compression. Under ordinary conditions the adiabatic con stants for a solid substance are practically the same as the cor responding isothermal constants.
where (l,771,n) are the direction cosines of the normal to the sur face-element dS. The quantity under the integral sign is now in terpreted to be the x-component of the surface traction acting across the surface element dS, while the quantity within brackets is regarded as the x-component of the stress across this element. Denoting this quantity by X, the preceding equation may be written in the form and there are similar equations involving the y and z-components YvZv. These may be regarded as three of the conditions of equilibrium of the surface tractions on the elements of the closed surface S. The other conditions for the equilibrium of these sur face tractions are obtained by taking moments about the axes of co-ordinates; they are It must be remembered that considerable differences are found in the elastic constants of different samples of nominally the same substance. The assumption that a metal is homogeneous and isotropic is not very accurate, for modern metallurgical studies have shown that metals which have been subjected to certain treatments may be composed of long narrow crystals which impart to a specimen of kind of fibrous structure. Much work has indeed been done recently on the strength and elastic properties of single crystals, and these are found to differ con siderably from those of a solid composed of an aggregate of small crystals and an amorphous substance which binds them to gether. The fibrous structure of wood is well known and several elastic constants are really needed to describe its properties, and these depend largely upon the amount of moisture in the wood. The value of E generally used corresponds to tension along the grain.