Elastic Energy and Material Stability

Imagine loading a small material element slowly, without heat exchange changing its elastic constants and without plastic flow. The incremental mechanical work per unit volume is the double contraction of stress with the strain increment. For a linear elastic material loaded from its unstressed reference state, stress grows linearly along a proportional loading path. Integrating therefore produces the familiar factor of one half. Energy density has units of pascals, equivalently joules per cubic metre; it is not the total energy of the whole specimen.

For small symmetric strain, an isotropic constitutive law uses two Lamé constants. Differentiating its quadratic energy with respect to strain recovers stress. Here repeated indices are summed, so off-diagonal tensor shear entries contribute twice in the contraction. Engineering shear strain is twice the corresponding tensor component. Mixing these conventions is a common source of an incorrect factor of two. The energy formula assumes a stress-free reference and linear response; an already highly stressed material requires a more careful incremental analysis.

Split the strain into its trace, which describes the first-order fractional volume change, and a traceless part that describes distortion. These parts are orthogonal under the tensor contraction. Substituting the split into the energy removes the coupling between them. The result says that a pure volume change is charged according to the bulk modulus, while a pure shape change is charged according to the shear modulus. Testing these two independent deformation families tests all symmetric small strains in this isotropic model.

Strict positive energy requires a positive shear modulus and a positive bulk modulus. It does not require the first Lamé constant to be positive. For example, a moderately negative first Lamé constant can coexist with both required positive moduli; such a material has an allowed auxetic elastic response. If either required modulus is zero, some nonzero strain costs no quadratic energy, so strict stability is lost. If either is negative, a strain in the corresponding family lowers the energy of the reference state.

Material stability is a local statement about the constitutive energy. A slender compressed column can buckle even when its material has positive shear and bulk moduli. The column gains lateral deflection and releases work from the compressive load; the total potential energy includes that external loading contribution as well as stored bending energy. Boundary supports and geometry then determine whether the complete structure is stable. A check on elastic constants cannot replace a structural buckling calculation.

Use the quadratic-function lab as a one-coordinate picture of energy curvature. Before changing anything, predict what happens to the restoring slope at a fixed positive position when the coefficient doubles. Check the tangent readout: for the displayed quadratic, the slope doubles. This illustrates positive curvature in one chosen direction. The lab plots a dimensionless scalar quadratic and its area, not the six-dimensional strain energy, a material specimen, or a buckling test. Our actual stability conclusion comes from the volumetric and deviatoric decomposition, not from that picture alone.