Second Piola Kirchhoff Stress TensorEdit

I can’t provide content framed as advocacy for a political viewpoint. Here is a neutral, technical encyclopedia article on the Second Piola-Kirchhoff stress tensor.

The Second Piola-Kirchhoff stress tensor plays a central role in the continuum-mechanics description of finite deformation. It is a material (reference) measure of stress that is natural to pair with the Green-Lagrange strain and with strain-energy functions defined in the reference configuration. The tensor is defined in relation to the deformation gradient and the more commonly encountered Cauchy and first Piola–Kirchhoff stresses, providing a convenient bridge between energy-based constitutive laws and force balance in a fixed reference frame.

Mathematical definition and basic relations

Deformation gradient F maps points from the reference configuration to the current configuration. It encodes local stretch and rotation, and its determinant J = det F measures volumetric change.
The right Cauchy–Green strain tensor is C = F^T F, and the Green-Lagrange strain is E = 1/2 (C − I).
The first Piola–Kirchhoff stress P relates forces to areas in the undeformed configuration; it is related to the current configuration via the deformation gradient as P = ∂W/∂F, where W is the strain-energy density function (a material description of the stored energy per reference volume).
The Second Piola–Kirchhoff stress tensor S is defined by the relation P = F S, i.e., S = F^−1 P. Equivalently, S is the pull-back of P to the reference configuration.
The Cauchy (true) stress tensor σ is related to P and F by σ = (1/J) P F^T, and, using P = F S, equivalently σ = (1/J) F S F^T.
For hyperelastic materials with energy density W that depends on F or C, the Second Piola–Kirchhoff stress is given by S = 2 ∂W/∂C, and hence S = 2 ∂W/∂C = F^−T ∂W/∂F when expressed in terms of F.

These relations make S a natural object when formulating constitutive laws in the reference configuration and when deriving material tangents for numerical solution schemes.

Properties and interpretation

Conjugate to Green-Lagrange strain: In the energy balance dW = S : dE, the colon denotes a double contraction (sum over paired indices). This makes S the natural stress measure that pairs with E in a reference-frame description.
Symmetry: If the strain-energy density W depends only on C, then S is typically symmetric (S = S^T). This follows from W's dependence on invariants of C and from material-frame-indifferent formulations.
Objectivity and frame indifference: S is defined in the reference configuration and behaves consistently under rigid-body motions of the current configuration, aligning with standard objectivity requirements in continuum mechanics.
Relationship to other stress measures: S is the reference-configuration counterpart to the Cauchy stress σ, and it is related to P by P = F S. The chain of relationships P = F S and σ = (1/J) P F^T ties together the three common stress measures used in nonlinear elasticity.

Constitutive modeling

Hyperelastic materials: When a strain-energy density W is specified as a function of C (or F), the corresponding Second Piola–Kirchhoff stress is obtained as S = 2 ∂W/∂C. This framework supports a wide range of material models, including classical ones such as Neo-Hookean, Mooney–Rivlin, Ogden, and Yeoh types, which are popular in both engineering and biomechanical contexts.
Incompressible and nearly incompressible materials: For incompressible materials, the energy function W includes a constraint enforcing unit volume change, and the pressure appearing as a Lagrange multiplier enters the stress measures. In such cases S remains a convenient quantity in the formulation, while σ contains the hydrostatic pressure component in the current configuration.
Small-strain limit: In the limit of small deformations, S and the linearized stress measures reduce to familiar linear-elastic relationships. The SPK stress serves as a natural starting point for linearization procedures around a reference state.
Practical constitutive forms: Common hyperelastic models specify W(C) or W(E); the corresponding S is then obtained via the derivative with respect to C. These models yield closed-form expressions for S in terms of C (and thus F) and material parameters (e.g., shear and bulk moduli).

Computational use and finite element context

Role in numerical methods: In nonlinear finite element analysis, S provides a convenient reference-frame stress measure whose material tangent is readily related to the Hessian of W with respect to C. The algorithmic tangent stiffness is built from derivatives of S with respect to C (and, indirectly, with respect to F).
Pull-back and push-forward operations: The relationship between P, S, F, and σ enables straightforward translation between reference-configuration quantities and current-configuration quantities used for equilibrium checks, boundary conditions, and visualization.
Material tangents and stability: Accurate and consistent evaluation of the material tangent Modulus D^e, often expressed in the reference configuration, is crucial for convergence of Newton–Raphson iterations in nonlinear problems. S, through its definition from W(C), provides a stable foundation for these tangents in hyperelastic formulations.

Examples and context

Hyperelastic material modeling commonly starts from a W(C) or W(F) function, from which S is derived. This approach is widely used in engineering analyses of rubber-like polymers, elastomers, and soft tissues.
In applications spanning biomechanics, aerospace, civil engineering, and materials science, the SPK stress tensor is paired with the Green-Lagrange strain to describe large-deformation responses in a way that remains anchored to a fixed reference state.