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Strain energy density function
Strain energy density function
Strain energy density function
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A strain energy density function or stored energy density function is a scalar-valued function that relates the strain energy density of a material to the deformation gradient.

Equivalently,

where is the (two-point) deformation gradient tensor, is the right Cauchy–Green deformation tensor, is the left Cauchy–Green deformation tensor,[1][2] and is the rotation tensor from the polar decomposition of .

For an anisotropic material, the strain energy density function depends implicitly on reference vectors or tensors (such as the initial orientation of fibers in a composite) that characterize internal material texture. The spatial representation, must further depend explicitly on the polar rotation tensor to provide sufficient information to convect the reference texture vectors or tensors into the spatial configuration.

For an isotropic material, consideration of the principle of material frame indifference leads to the conclusion that the strain energy density function depends only on the invariants of (or, equivalently, the invariants of since both have the same eigenvalues). In other words, the strain energy density function can be expressed uniquely in terms of the principal stretches or in terms of the invariants of the left Cauchy–Green deformation tensor or right Cauchy–Green deformation tensor and we have:

For isotropic materials,

with

For linear isotropic materials undergoing small strains, the strain energy density function specializes to

[3]

A strain energy density function is used to define a hyperelastic material by postulating that the stress in the material can be obtained by taking the derivative of with respect to the strain. For an isotropic hyperelastic material, the function relates the energy stored in an elastic material, and thus the stress–strain relationship, only to the three strain (elongation) components, thus disregarding the deformation history, heat dissipation, stress relaxation etc.

For isothermal elastic processes, the strain energy density function relates to the specific Helmholtz free energy function ,[4]

For isentropic elastic processes, the strain energy density function relates to the internal energy function ,

Examples

[edit]

Some examples of hyperelastic constitutive equations are:[5]

See also

[edit]
Wikiversity has learning resources about Continuum mechanics/Thermoelasticity

References

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Strain energy density function

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from Grokipedia
The strain energy density function, often denoted as WW or UU, is a scalar-valued function in that quantifies the stored per unit reference volume in a material subjected to deformation. It serves as the foundational constitutive relation for hyperelastic materials, enabling the derivation of stress tensors—such as the second Piola-Kirchhoff stress S=2∂W∂CS = 2 \frac{\partial W}{\partial \mathbf{C}}—directly from its partial derivatives with respect to kinematic measures like the right Cauchy-Green deformation tensor C=FTF\mathbf{C} = \mathbf{F}^T \mathbf{F}, where F\mathbf{F} is the deformation gradient. In hyperelasticity, the function WW is typically formulated to depend on the principal invariants I1,I2,I3I_1, I_2, I_3 of C\mathbf{C} for isotropic materials, ensuring path-independent loading-unloading behavior and thermodynamic consistency under isothermal conditions, as dictated by the Clausius-Duhem inequality. For nearly incompressible materials, volumetric terms are incorporated, such as W=W~(C)+U(J)W = \tilde{W}(\mathbf{C}) + U(J), where J=det⁡FJ = \det \mathbf{F} and U(J)U(J) penalizes deviations from incompressibility. This framework extends , where WW reduces to a W=12ε:C:εW = \frac{1}{2} \boldsymbol{\varepsilon} : \mathbb{C} : \boldsymbol{\varepsilon} in terms of the infinitesimal strain tensor ε\boldsymbol{\varepsilon} and the C\mathbb{C}. Prominent models include the Neo-Hookean function W=μ2(I1−3)+12K(J−1)2W = \frac{\mu}{2} (I_1 - 3) + \frac{1}{2} K (J - 1)^2, which captures Gaussian chain statistics in networks like rubber, and the Mooney-Rivlin extension W=C1(I1−3)+C2(I2−3)+12K(J−1)2W = C_1 (I_1 - 3) + C_2 (I_2 - 3) + \frac{1}{2} K (J - 1)^2, accounting for additional nonlinearities in moderate strains. These functions are widely applied in simulating large deformations in biological tissues, elastomers, and engineering components, with parameters calibrated via experimental stress-strain data.

Fundamentals

Definition

The strain energy density function, often denoted as WW, is a scalar-valued function that quantifies the elastic strain energy stored per unit reference volume of the material body. It relates the stored energy to measures of deformation, typically expressed as W=W(ε)W = W(\varepsilon) for infinitesimal strains, where ε\varepsilon is the strain tensor, or W=W(F)W = W(\mathbf{F}) for finite deformations, where F\mathbf{F} is the deformation gradient tensor. In finite deformations, this is per unit reference volume to account for the deformation gradient. This concept was introduced by George Green in 1839 as part of his foundational work on the mathematical theory of elasticity, where he proposed the as a of strain components to describe the in elastic solids. Green's approach utilized the principle of to express the internal forces as the differential of a function dependent on the strains, revolutionizing the formulation of elastic constitutive relations. In SI units, the strain energy density has dimensions of per unit , measured in joules per cubic meter (J/m³). The total elastic UU stored in a material body is obtained by integrating the strain energy density over the body's reference : U=∫VW dVU = \int_V W \, dV. This provides the overall associated with the deformation, serving as a basis for variational principles in elasticity.

Physical Interpretation

The strain energy density function, often denoted as WW, quantifies the elastic potential stored per unit reference volume within a deformable as a result of mechanical deformation. It embodies the work expended by internal forces to achieve reversible straining, which is subsequently released as the material returns to its undeformed configuration in purely elastic responses. This stored arises from the reconfiguration of atomic or molecular bonds under load, maintaining without permanent alteration to the material's structure. In hyperelastic materials, WW functions as a , contingent exclusively on the instantaneous deformation tensor and independent of the deformation history or loading trajectory. This path independence ensures that the total stored energy remains consistent for any sequence of deformations leading to the same final state, underpinning the conservative nature of hyperelastic behavior. Such a property facilitates the modeling of materials that exhibit fully recoverable deformations under arbitrary loading paths. Conceptually, [W](/page/W)[W](/page/W) behaves like the of a coiled spring for strains, where accumulates quadratically with increasing deformation, reflecting a linear stress-strain relationship. However, under finite strains typical of large deformations, this accumulation turns nonlinear, allowing [W](/page/W)[W](/page/W) to capture phenomena such as stiffening or softening that deviate from simple storage. This transition highlights the function's adaptability to both modest elastic distortions and extreme material excursions. Distinct from , which stems from the motion of material particles, WW is exclusively potential in character, tied to static configurational changes in elastic solids and inherently free of dissipative losses like those in plastic flow or viscous . This elastic exclusivity positions WW as a for analyzing reversible cycles in materials devoid of .

Mathematical Formulation

Infinitesimal Strain Case

In the infinitesimal strain case, the strain energy density function is developed under the assumptions of small deformations in theory, where the magnitude of the strain tensor satisfies |ε| << 1, allowing the neglect of higher-order terms in the expansion of the deformation gradient. This approximation simplifies the kinematics and constitutive relations, enabling a direct connection to classical mechanics principles. The appropriate strain measure is the infinitesimal (or Cauchy) strain tensor, defined as ε=12(∇u+(∇u)T),\boldsymbol{\varepsilon} = \frac{1}{2} \left( \nabla \mathbf{u} + (\nabla \mathbf{u})^T \right), where u\mathbf{u} is the displacement field and ∇\nabla denotes the gradient operator. This symmetric second-order tensor captures the linear approximation of the deformation, focusing on changes in length and angle without considering rotations explicitly. For hyperelastic materials in this regime, the strain energy density function takes a quadratic form W(ε)=12ε:C:ε,W(\boldsymbol{\varepsilon}) = \frac{1}{2} \boldsymbol{\varepsilon} : \mathbf{C} : \boldsymbol{\varepsilon}, where C\mathbf{C} is the fourth-order elasticity (stiffness) tensor, and :: denotes the double contraction (a scalar product between tensors). This expression represents the elastic potential energy stored per unit volume due to deformation, assuming path-independent loading and reversible behavior. The corresponding Cauchy stress tensor σ\boldsymbol{\sigma} is derived from the potential as the partial derivative σ=∂W∂ε=C:ε,\boldsymbol{\sigma} = \frac{\partial W}{\partial \boldsymbol{\varepsilon}} = \mathbf{C} : \boldsymbol{\varepsilon}, which establishes the linear relation central to Hooke's law, linking stress directly to strain through the material's elastic properties encoded in C\mathbf{C}. This formulation ensures that the stress vanishes when the strain is zero and supports the superposition principle for small perturbations.

Finite Strain Case

In the finite strain case, which addresses large deformations in nonlinear elasticity, the strain energy density function WW is defined per unit volume in the reference configuration and serves as the core of hyperelastic constitutive models. This formulation accounts for geometric nonlinearities where traditional small-strain approximations fail, using objective measures that remain invariant under rigid body motions. The deformation gradient tensor F=∂x∂X\mathbf{F} = \frac{\partial \mathbf{x}}{\partial \mathbf{X}} describes the mapping from the reference configuration position X\mathbf{X} to the current configuration position x\mathbf{x}, with the assumption det⁡(F)>0\det(\mathbf{F}) > 0 ensuring local invertibility and preservation of material orientation. Common objective strain measures derived from F\mathbf{F} include the right Cauchy-Green deformation tensor C=FTF\mathbf{C} = \mathbf{F}^T \mathbf{F} and the Green-Lagrange strain tensor E=12(C−I)\mathbf{E} = \frac{1}{2} (\mathbf{C} - \mathbf{I}), where I\mathbf{I} is the identity tensor. The strain energy density WW is typically expressed as a scalar function of these measures, such as W=W(F)W = W(\mathbf{F}) or W=W(E)W = W(\mathbf{E}), capturing the stored without dependence on the deformation path for reversible processes. The stresses conjugate to these kinematic quantities are obtained through hyperelasticity relations: the first Piola-Kirchhoff stress tensor P=∂W∂F\mathbf{P} = \frac{\partial W}{\partial \mathbf{F}} relates nominal stress in the reference frame, while the second Piola-Kirchhoff stress tensor S=∂W∂E=F−1P\mathbf{S} = \frac{\partial W}{\partial \mathbf{E}} = \mathbf{F}^{-1} \mathbf{P} provides a symmetric, objective measure pulled back to the reference configuration. A fundamental requirement is the principle of material objectivity (or frame-indifference), stipulating that W(QF)=W(F)W(\mathbf{Q} \mathbf{F}) = W(\mathbf{F}) for any proper orthogonal tensor Q\mathbf{Q} (i.e., QTQ=I\mathbf{Q}^T \mathbf{Q} = \mathbf{I}, det⁡(Q)=1\det(\mathbf{Q}) = 1), ensuring the energy is unaffected by superimposed rigid rotations. In the limit of small deformations where ∥F−I∥→0\|\mathbf{F} - \mathbf{I}\| \to 0, this finite strain framework reduces to the infinitesimal strain case.

Specific Models

Linear Elastic Materials

In linear elastic materials, the strain energy density function describes the stored per unit under small deformations, assuming a quadratic relationship between stress and strain. For isotropic materials, which exhibit uniform properties in all directions, the strain energy density WW is expressed as W=λ2(tr⁡ε)2+με:ε,W = \frac{\lambda}{2} (\operatorname{tr} \boldsymbol{\varepsilon})^2 + \mu \boldsymbol{\varepsilon} : \boldsymbol{\varepsilon}, where ε\boldsymbol{\varepsilon} is the infinitesimal strain tensor, tr⁡ε\operatorname{tr} \boldsymbol{\varepsilon} is its trace (volumetric strain), :: denotes the double contraction, λ\lambda is the first Lamé constant (related to ), and μ\mu is the second Lamé constant (). These Lamé constants connect to more commonly used parameters: EE (stiffness under uniaxial tension) and ν\nu (lateral contraction ratio), via λ=Eν(1+ν)(1−2ν),μ=E2(1+ν).\lambda = \frac{E \nu}{(1 + \nu)(1 - 2\nu)}, \quad \mu = \frac{E}{2(1 + \nu)}. This form arises from the assumption of hyperelasticity in the linear regime, ensuring path-independent energy storage and symmetry in the stress-strain response. For anisotropic linear elastic materials, where properties vary with direction (as in or composites), the strain energy density takes a more general : W=12ε:C:ε,W = \frac{1}{2} \boldsymbol{\varepsilon} : \mathbb{C} : \boldsymbol{\varepsilon}, with C\mathbb{C} the fourth-order tensor relating stress σ=C:ε\boldsymbol{\sigma} = \mathbb{C} : \boldsymbol{\varepsilon}. The tensor C\mathbb{C} has 81 components in general, but symmetries of stress and strain (major and minor) reduce the number of independent constants to 21 for triclinic symmetry (lowest crystal symmetry, no rotational invariance). Higher symmetries further reduce this: for example, monoclinic symmetry (one mirror plane) yields 13 independent constants. To simplify computations, contracts the tensor into a 6×6 C\mathbf{C}, mapping the six unique strain components (three normal, three shear) to a vector, facilitating numerical implementation in analyses. The total strain energy U=∫VW dVU = \int_V W \, dV over the body volume VV governs equilibrium through the principle of minimum , where the true displacement field minimizes UU subject to boundary conditions, subject to external loads. Stationarity condition δU=0\delta U = 0 (first variation) yields the Navier equations of equilibrium in displacement form for isotropic cases: μ∇2u+(λ+μ)∇(∇⋅u)+f=0,\mu \nabla^2 \mathbf{u} + (\lambda + \mu) \nabla (\nabla \cdot \mathbf{u}) + \mathbf{f} = 0, with u\mathbf{u} the displacement vector and f\mathbf{f} body forces; analogous forms hold for anisotropic media via component-wise . This linear framework assumes strains and instantaneous elastic recovery, but it fails to capture behaviors under large deformations (where geometric nonlinearities dominate) or in viscoelastic materials (exhibiting time-dependent recovery).

Hyperelastic Materials

Hyperelastic materials are characterized by a constitutive response where the stress is derived from the of a scalar density function WW with respect to the appropriate strain measure, ensuring a path-independent and conservative deformation process. In the context of finite strains, this framework posits that the material stores all work done during deformation as recoverable , with the obtained from partial derivatives of WW with respect to the invariants of the deformation tensor, as developed in the general theory of large elastic deformations. For many hyperelastic materials, such as rubbers, an incompressible assumption is often adopted, where the strain energy density function depends on the first two principal invariants of the right Cauchy-Green deformation tensor C\mathbf{C}, denoted W=W(I1,I2)W = W(I_1, I_2), with I1=tr⁡CI_1 = \operatorname{tr} \mathbf{C} and I2=12((tr⁡C)2−tr⁡(C2))I_2 = \frac{1}{2} \left( (\operatorname{tr} \mathbf{C})^2 - \operatorname{tr} (\mathbf{C}^2) \right). Volume preservation is enforced by the condition det⁡F=1\det \mathbf{F} = 1, where F\mathbf{F} is the deformation gradient, leading to an additional hydrostatic pressure term pp in the stress expression, such that the Cauchy stress is σ=−pI+2∂W∂I1B−2∂W∂I2B−1\boldsymbol{\sigma} = -p \mathbf{I} + 2 \frac{\partial W}{\partial I_1} \mathbf{B} - 2 \frac{\partial W}{\partial I_2} \mathbf{B}^{-1}, with B=FFT\mathbf{B} = \mathbf{F} \mathbf{F}^T the left Cauchy-Green tensor. A foundational phenomenological model is the Neo-Hookean form, which for compressible materials takes the strain energy density as W=μ2(I1−3)+f(J),W = \frac{\mu}{2} (I_1 - 3) + f(J), where μ>0\mu > 0 is the shear modulus, J=det⁡FJ = \det \mathbf{F}, and f(J)f(J) accounts for volumetric changes, often chosen as f(J)=κ2(J−1)2f(J) = \frac{\kappa}{2} (J - 1)^2 with bulk modulus κ≫μ\kappa \gg \mu to approximate near-incompressibility. This model extends the linear elastic response to moderate finite strains and is widely used for soft tissues and elastomers due to its simplicity and grounding in statistical mechanics of polymer networks. The Mooney-Rivlin model generalizes the Neo-Hookean by incorporating the second invariant, with the incompressible density given by W=C10(I1−3)+C01(I2−3),W = C_{10} (I_1 - 3) + C_{01} (I_2 - 3), where C10C_{10} and C01C_{01} are material constants related to the initial by μ=2(C10+C01)\mu = 2 (C_{10} + C_{01}). This form captures the stiffening behavior observed in rubbers under biaxial loading better than the Neo-Hookean, as validated through experiments on vulcanized rubber. For broader applicability to large deformations, the Ogden model expresses the strain energy in terms of the principal stretches λi\lambda_i (eigenvalues of V=B\mathbf{V} = \sqrt{\mathbf{B}}
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