The Elasticity Tensor

The constants s and c ( = 1 /s) are known as the elastic compliance constant and the elastic stiffness constant, respectively. The elastic stiffness constant is the elastic modulus, which is seen to be the ratio of stress to strain. In the case of normal stress-normal strain (Fig. 10.3a) the ratio is the Young s modulus, whereas for shear stress-shear strain the ratio is called the rigidity, or shear, modulus (Fig. 10.36). The Young s modulus and rigidity modulus are the slopes of the stress-strain curves and for nonHookean bodies they may be defined alternatively as da-/ds. They are requited to be positive quantities. Note that the higher the strain, for a given stress, the lower the modulus. 

Think about what happens when, say, an elastomer is under tensile stress. The elastic constants, s and c, cannot be scalar quantities, otherwise Eqs. 10.5 and 10.6 would not completely describe the elastic response. When the elastomer is stretched, a contraction 

Lia ure 10.3. The different forces acting in tensile deformation and shear deformation. 

Tensile and shear forces are not the only types of loads that can result in deformation. Compressive forces may as well. For example, if a body is subjected to hydrostatic pressure, which exists at any place in a body of fluid (e.g. air, water) owing to the weight of the fluid above, the elastic response of the body would be a change in volume, but not shape. This behavior is quantified by the bulk modulus, B, which is the resistance to volume change, or the specific incompressibihty, of a material. A related, but not identical property, is hardness, H, which is defined as the resistance offered by a material to external mechanical action (plastic deformation). A material may have a high bulk modulus but low hardness (tungsten carbide, B = 439 GPa, hardness = 30 GPa). 

Because stress and strain are vectors (first-rank tensors), the forms of Eqs. 10.5 and 10.6 state that the elastic constants that relate stress to strain must be fourth-rank tensors. In general, an wth-rank tensor property in p dimensional space requires p coefficients. Thus, the elastic stiffness constant is comprised of 81 (3 ) elastic stiffness coefficients, 

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Fig. 11. Tensor-valued elasticity parameters in a human breast in vivo. A dotted circle symbolizes a carcinoma previously localized using gadolinium-enhanced Ti-weighted imaging. Eigenvalues Ei, E2, and E3 of the elasticity tensor are shown in (a), (b), and (c) respectively. Also shown in (d) is the isotropic elasticity...

Using relations (2.5) and (2.6) we can determine the elasticity tensor which describes the linear relation between components of the stress and strain tensors. 2 slr.ss = CEstta n is therefore an expression of Hooke s law for anisotropic crystals... 

Example 15.2-2 Determine the non-zero elements of the elasticity tensor ciJki for a crystal of D4 symmetry. The generalized form of Hooke s law is... 

Cowin, S.C. 1985. The relationship between the elasticity tensor and the fabric tensor. Mech. Mater. 4, 137-147. 

Given the elastic tensor we can obtain Reuss average, isotropic bulk and shear moduli... 

These simplifications reduce the size of the elasticity tensors from [9 x 9] to [6 x 6], with 36 elastic coefficients. The shorthand notation normally used for the elasticity tensors are now introduced, namely, that the subscripts become 1 11 2 22 3 33 4 23, 32 5 31, 13 and 6 12, 21. With this change, the elastic stiffness tensor may be written in matrix form as ... 

It is important to use the exact strain tensor definition, Eq. (6), to achieve rotational invariance with respect to lattice rotation the conventional linear strain tensor only provides differential rotational invariance of u in Eq. (7).hierarchy of approximations may be used for the elastic tensor 7. The most rigorous approach is to transform the bulk elastic tensor c according to... 

We have learned already that any elastic oscillation can be represented as a superposition of the compressional and shear waves, which correspond to the potential and solenoidal parts of the elastic displacement field. Therefore, it is clear that the elastic tensor G can also be represented as the sum of the potential and solenoidal components, described by tensor functions G W and G respectively ... 

The coupling of the mechanical field with the chemical field is realized as follows As there are bound charges present in the gel, a jump in the concentrations of the mobile ions at the interface between the gel and the solution is obtained. This difference in the concentrations leads to an osmotic pressure difference Ati between gel and solution. As a consequence of this pressure difference, the gel takes up solvent, which leads to a change of the swelling of the gel. This deformation is described by the prescribed strain e. This means that the mechanical stress is obtained by the product of the elasticity tensor C and the difference of the total (geometrical) strain e and the prescribed strain ... 

The fourth order tensor C is the derivative of the elastic tensor with respect to damage variable ... 

The elasticity tensor transforms according to equation (4.7), i.e. like the product of four coordinates. An inversion center transforms the coordinates (xj,X2,X3) into (x i,x 2,x3) = (-xi,-X2,-X3), hence = (-= S npq- Thus the tensor is invariant with respect to inversion. 

The relation between e and d may be calculated by using the elasticity tensor which expresses as a function of [Pg.195]

For both magnesite and calcite, the elastic bulk modulus Bq was computed straightforwardly by the Murnaghan interpolation formula, while of the elasticity tensor only the C33 component and the C + C 2 linear combination could be calculated in a simple way. The relations used are C = (l/Vo)c (d L /crystal structure. To derive other elastic constants, the symmetry must be lowered with a consequent need of complex calculations for structural relaxation. A detailed account of how to compute the Ml tensor of crystal elasticity by use of simple lattice strains and structure relaxation was given previously[10, 11]. For the present deformations only the c-o ( ) relaxation need be considered. The results are reported in Table 6, together with the corresponding values extrapolated to 0 K from experimental data (Table 2). For calcite, the mea-... 

They provide a full description of the mechanical properties of crystalline materials. B is related to the elastic tensor. In the case of a cubic system, where only three independent components of the elastic tensor differ from zero, B can be obtained from Cn and Cu as... 

Here, the right part of the equation is abbreviation of the middle part suggested by Einstein since symbols k and / appear twice as suffixes at Kyu and Cki-, we may remember this and remove a bulky symbol of the sum. By this convention, we always must make a summation over the repeated suffixes. The elasticity tensor KijM is a fourth rank tensor with 9 x 9 = 81 components (81 mathematically possible elastic moduli ). However, even in crystals of the lowest symmetry (the triclinic system) due to physical equivalence of = KjM= Kijik = Kkuj, a number of moduli reduces to 21. 

Here, E denotes the elasticity tensor of the material. Examples of constitutive laws for smart materials are given in the subsequent section. 

Since the differentiation of (3.111) and (3.112) is commutative, the elastic tensor A and compliance tensor B show symmetry ... 

The elasticity tensor (7 is a tensor of fourth order. It can be considered as a four-dimensional matrix with three components in each of its 4 directions. Its 3 = 81 components Cjjfc are the material parameters that completely describe the (linear) elastic behaviour. 

Because the stress and the strain tensor contain only 6 independent components each, due to their symmetry, the elasticity tensor C needs only 6 = 36 independent parameters. 

That not all 81 components of the elasticity tensor are needed can be most easily understood using an example. For <712, we find from equation (2.20)... 

Furthermore, because <712 = <721, we can also set Cijki = Cjiki- The two S5rmmetry conditions Cijki = Cjiki and Cijki = Cijik reduce the number of independent components of the elasticity tensor to 36. 

The elasticity tensor possesses further symmetries due to the existence of an elastic potential [108]. The elasticity matrix (C0/3) is symmetric because of this and the number of independent components reduces further to 21 (6 diagonal and 15 off-diagonal ones). 

The elasticity tensor is thus the derivative of the stress with respect to the strain. 

Because the sequence of taking the derivatives is arbitrary, we find the symmetry condition Cijki = CkUj for the elasticity tensor, or, for the elasticity matrix, (Cap) = (C a). 

A material is mechanically isotropic if all of its mechanical properties are the same in all spatial directions. The elasticity tensor must thus remain unchanged by arbitrary rotations of the material or the coordinate system. Its components must be invariant with respect to rotations. 

This corresponds to pure shear with 712 = 2s, see figure 2.8(b). If we ignore the isotropy of the elasticity tensor for a moment, we have to assume that its components are different in different coordinate systems. 

S is the compliance tensor, the inverse of the elasticity tensor C Because in-verting a matrix is an awkward calculation, the components of the compliance matrix are written explicitly here ... 

Unspecified components vanish. Thus, the three independent constants An, Ai2, and A44 remain. If the coordinate system is not parallel to the edges of the unit cell, a coordinate transformation of the elasticity tensor has to be used to find the components. In this case, the elasticity matrix takes a shape different from that in equation (2.34). 

The considerations of the previous sections dealt with the two most extreme load cases If loaded in fibre direction, the stiffening effect of the fibres is maximal, if loaded perpendicularly, it is minimal. Under arbitrary loads, it is necessary to calculate the components of the elasticity tensor (see section 2.4.2). Depending on the fibre arrangement, couplings between normal and shear components can occur. For example, this can be exploited to construct aerofoils that twist on bending, with normal stresses causing shear strains. 

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