A hyperelastic material derives its elastic stress from a strain energy density. In reference-volume units its first Piola-Kirchhoff stress tensor is the derivative of that density with respect to the deformation gradient.
A hyperelastic energy well is a set of deformation gradients minimizing the stored-energy function at a fixed temperature. Material frame indifference makes each well invariant under left multiplication by proper rotations. A phase-changing crystal can have a single austenitic rotational well at high temperature and several martensitic wells below a transition. Continuous deformations cannot freely join arbitrary wells: planar interfaces require a rank-one connection between the selected gradients.
On positive-determinant deformation gradients, the first Piola-Kirchhoff stress is . Its zero set is : multiplication by gives , and determinants give . These are also the global minimizers, since and has its unique minimum at . The minimum energy is rather than zero; an additive constant changes no stress.
An unconstrained natural state is a stress-free homogeneous deformation gradient. For , its equivalent conditions are and . An interior differentiable minimum of the strain energy density is a natural state, although a stationary state need not minimize the energy. Constraints require allowing their reaction stress; the unconstrained equivalence must then be modified. Under strict Baker-Ericksen inequalities, an isotropic natural state is a dilation times a proper rotation.
For a material isotropy and material frame indifference strain energy density , take , , , , . The chain rule and give . Multiplying by yields the displayed Cauchy stress tensor. Material isotropy allows an identity term and a quadratic term in , as well as the term proportional to .
A hyperelastic material is homogeneous when its strain energy density has no explicit dependence on the reference configuration position. Its first Piola-Kirchhoff stress tensor is . Homogeneity of the material does not require the deformation gradient to be spatially constant.
For an isotropic hyperelastic material with incompressibility, the principal stresses are . This follows by evaluating the Cauchy stress tensor formula in principal directions. For the reduced strain energy density , the chain rule and equality of the tangential derivatives give .
For a plane material-coordinate elastic wave with , set . The longitudinal equation is . Arbitrary nonuniform pure transverse profiles thus require the displayed constitutive compatibility, which does not follow from isotropy alone. If it holds and , the shear equation closes as with speed .
An explicit objective isotropic example is , with , , and , . On , and , so at while the shear nominal stress is the displayed cubic. The natural-state quadratic energy has Lamé parameters , and the shear speed is .
This objective isotropic constitutive model uses the Green-Lagrange strain tensor . It reproduces linear elasticity at small strain. Under simple shear , its longitudinal nominal stress is . Nonuniform finite shear therefore generates longitudinal acceleration, even though the corresponding infinitesimal transverse mode is uncoupled.
At a stress-free natural reference state, an objective isotropic hyperelastic material with energy per unit mass has , where . Twice differentiating gives the displayed Lamé parameters and the usual elastic stiffness tensor. A prestressed reference requires initial-stress corrections.
At leading thin-shell order, a spherical balloon with reference radius , thickness and current radius has reference elastic energy . Equating its differential to pressure work gives the displayed pressure difference. Its thickness becomes by incompressibility. The same result follows from the spherical membrane force balance , with current tangential Cauchy stress tensor component and the equibiaxial nominal tension of an incompressible sheet. Thin-shell validity requires .
For unconstrained hyperelastic material, the Hessian of the strain energy density gives . Commuting the second derivatives gives major symmetry . These material-coordinate moduli linearize about a finite base deformation; they need not have the minor symmetries of a small-strain elastic stiffness tensor. Material constraints require their additional reaction increments and admissibility equations.
For a static base state with zero incremental body force, homogeneous incremental tractions or homogeneous Dirichlet boundary conditions eliminate the boundary term in integration by parts. The displayed Rayleigh quotient then gives real squared frequencies because of major symmetry. Positive moduli give strictly positive squared frequencies for modes with nonzero displacement gradient. Pure dead loading leaves constant translations at zero; fixed displacement conditions remove those modes.
Strong ellipticity requires the displayed inequality for every nonzero displacement vector and material direction . Equivalently, every acoustic tensor is positive-definite. It is weaker than positivity of the moduli on all matrices. A quadratic minor null Lagrangian vanishes on rank-one matrices, so adding such a term does not change the acoustic tensor or this condition.
The acoustic tensor determines the plane waves of a locally uniform incremental elastic medium. For a unit material direction , its eigenvalues are and its eigenvectors are displacement polarizations. Major symmetry of the incremental elastic moduli makes a real symmetric matrix. Positivity on every rank-one matrix makes all its wave speeds real and nonzero.
The elastic stiffness satisfies for every nonzero polarization and propagation vector . It makes the elastic acoustic tensor positive definite, ensuring positive squared wave speeds. Positive strain energy is a sufficient stronger condition in ordinary linear elasticity.
For a regular material constraint , admissible rates of the deformation gradient lie in the kernel of . The difference between the nominal stress and the derivative of the strain energy density annihilates that tangent space by virtual work, so it is a Lagrange multiplier times . For incompressibility, the derivative of the determinant is its cofactor matrix, and the material-first reaction is when .
A thin isotropic hyperelastic material subject to incompressibility, with equal in-plane principal stretches has transverse stretch . With unloaded transverse faces, . Hence , using isotropy to set . The derivative along the constrained path is , proving the displayed formula. The two equal nominal tensions do work per reference volume, exactly the change in strain energy density.
An incompressible Mooney-Rivlin solid has strain energy density , with right Cauchy-Green deformation tensor , and when . It extends the neo-Hookean solid by a second invariant. A convention means , ; the minus sign must be retained when comparing material constants.
For incompressible simple shear with Mooney-Rivlin energy , zero traction on the third-coordinate faces fixes the pressure reaction. The remaining Cauchy stress has , and . The material face has current unit normal and tangent . Resolving gives and . Since , the traction-normal angle satisfies , proving the displayed formula. Consequently its two-sided small-shear slope is . The reference normal cannot replace the transformed normal in this calculation.
For an incompressible neo-Hookean solid, a convenient strain energy density is , where is the right Cauchy-Green deformation tensor. An additive constant in does not affect the stress. The shear modulus fixes the small-strain stiffness.
Consider an incompressible neo-Hookean prism with energy , zero nominal end traction normal to , and equal nominal lateral tractions on side faces whose normals enforce both in-plane principal stresses. Within positive diagonal homogeneous deformations the first Piola-Kirchhoff stress tensor has entries . Equilibrium makes constant; end freedom gives .
The function is strictly increasing on positive , so equal lateral stresses require . The determinant constraint then gives and the displayed load relation. Its derivative is , with limits minus infinity and infinity at the two ends of the positive axis. Thus there is exactly one positive diagonal equilibrium for every signed load. This uniqueness is confined to the homogeneous diagonal class and does not rule out other modes or nonhomogeneous equilibria under a general dead-loading problem.
For the volume-preserving family , equal dead nominal tension gives potentialIts derivative factors as . The nontrivial branch has a fold at , , and meets the trivial branch at . Stability in this one-parameter family is determined by , not by branch count alone. At there is only one nontrivial positive solution, ; the other branch value is .
In three-dimensional isotropic linear elasticity, decompose the symmetric strain as with and . The stored energy is . Its nonnegativity for every strain is equivalent to the displayed inequalities: the traceless and purely volumetric test strains prove necessity, and the sum of squares proves sufficiency.
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Hyperelastic materials, also known as Green elastic materials, are a class of materials that exhibit elastic behavior over large strains. They are characterized by a strain energy density function that describes how the material deforms under stress. Unlike linear elastic materials, which only return to their original shape after small deformations (typically under 5% strain), hyperelastic materials can undergo large strains and still return to their original configuration when the load is removed.