2.2.2. Major Symmetry

This is another symmetry present in the stiffness tensor which comes from energy considerations. It says C = C (15)

Major symmetry in the stiffness tensor indicates that the stiffness properties of a material do not change even if we exchange the indices associated with the stress and strain components. This is significant in understanding how materials respond elastically under different loads, and highlights the interconnectedness of stress and strain across the entire material.

Imagine a perfectly balanced seesaw where both sides are equally strong. If you push down on one side, both sides respond symmetrically - that’s like the major symmetry in materials. The way stress affects strain on one side reflects how strain affects stress on the other.

If the elongation of the spring is x and the spring constant is k, the force F that gets generated in the spring is given by F = kx (16) while the energy stored in the spring is given by (17)

This analogy illustrates how materials can be thought of like springs in a system. When a force is applied to a spring (representing stress), it elongates (representing strain). The energy stored in the spring due to this elongation is analogous to the strain energy stored in a material when it deforms under load. The relationship defined by F = kx shows how changes in stress (force) lead to strain (deformation), akin to how stiffness affects the energy stored in springs during deformation.

Think about stretching a rubber band. The more you stretch it (application of force), the more it deforms. Similarly, the additional energy (strain energy) stored in the rubber band when it's stretched illustrates the concept of how materials store energy when stressed.

We can also say that the energy is stored when a body gets deformed which comes from the work done on the body by the external load.

This point emphasizes that deformation of materials under load stores energy. The external forces perform work on the structure, and this work translates into potential energy held within the material’s elastic structure. This stored energy can be released when the load is removed, a critical aspect in applications like structural engineering and material design.

Consider a compressed spring in a toy. When squeezed, the toy stores energy. When released, that energy converts back into motion as the spring pushes outwards. Similarly, materials that deform under loads can store energy in this way, potentially causing reactions when the load is removed.

To get the total energy stored in a body, we can integrate the strain energy density over the entire volume of the body. Comparing with spring energy, we can write the strain energy density E as (19)

Strain energy density quantifies how much energy is stored per unit volume of material. By integrating this density over the body's volume, we calculate the total energy stored. This mathematical approach allows for accurate predictions of material behavior under various loading conditions and is essential for engineering design.

Imagine filling a balloon with air. The energy required to inflate it represents the work done on the balloon, similar to how strain energy is calculated from the density of stored energy in materials. Each part of the balloon holds a tiny portion of the total energy, which can be felt when the balloon pops, like material failure when the stress exceeds the capacity.

This analysis proves that although equation (11) has a general C tensor, we can always work with an equivalent Ɯ which has both major and minor symmetries.

This segment synthesizes the information about how the stiffness tensor can be simplified using the symmetries derived earlier. Understanding that the tensor can be represented in a more manageable form aids engineers in their calculations and theoretical models. By using matrices that respect these symmetries, one reduces the complexity involved in analyzing material behavior.

Think of organizing a messy desk. By rearranging books into labeled folders (the equivalent representation of the stiffness tensor), you can find the information more efficiently, similar to how this simplification helps engineers analyze material properties swiftly using the derived symmetries.

Due to major symmetry, the number of independent constants in C will further reduce. We can think of a 6×6 matrix with the rows representing the values of C corresponding to independent combinations of (i,j) and the columns representing the values of C corresponding to independent combinations of (k,l):

The reduction in independent constants stems from the major symmetry property, allowing us to systematically simplify the analysis of the stiffness tensor. This results in a structured matrix containing only unique combinations vital for defining material responses under stress, aiding both theoretical studies and practical applications in engineering.

Consider a family tree where you only keep track of unique relationships. Reducing the number of relationships to just the unique connections simplifies understanding the family dynamics, much like how reducing the constants simplifies studying material properties under load.

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