An approach to stress conversion between different deformation states for incompressible first‐invariant hyperelastic models

To characterize generally valid hyperelastic material parameters, tensile tests at different deformation states are essential. This necessity arises from the hyperelasticity theory for incompressible materials, which is based on the first and second invariants of the right Cauchy–Green tensor. These invariants describe the elongation and surface change during a deformation process. By comparing different deformation states using the incompressibility plane of invariants, an equivalent deformation state can be determined for first‐invariant hyperelastic models. This deformation equivalence results in equal strain energy and is used to derive a method for converting stress‐strain relations of different deformation modes. The presented approach is verified and validated with experimental data.


INTRODUCTION
For the design of technical components, engineers need an adequate material model to estimate the stress-strain relations in the component.In general, tensile tests provide the data basis for the selection and adaptation of material models.In case of rubber-like materials, a distinction is made between uniaxial, equibiaxial, and plane-strain tensile tests.Following the literature [1][2][3], these different deformation modes are required to determine material parameters, which are reliable over a wide range of deformation states.Material parameters based only on uniaxial test data are assumed to be error-prone in their material prediction.However, performing tests for different deformation states is a complex and time-consuming process that requires custom built measurement equipment and is therefore often omitted.
Assuming an elastic strain-energy density based on the first invariant, a method for converting the stress-strain behavior between deformation states is presented here.In detail, the uniaxial test data are used as reference to reformulate the equibiaxial and the plane-strain test data.
The stress conversion approach is obtained by formulating a deformation equivalence in Section 2 based on nonlinear kinematics.The considered uniaxial, equibiaxial, and plane-strain deformation states are compared using the incompressibility plane of invariants.By linking the nonlinear kinematic with the fundamentals of isotropic incompressible hyperelastic material models, the conversion approach is derived in Section 3. The verification and validation in Section 4 is based on hyperelastic models and experimental data for carbon black filled rubbers.Finally, the implications of these new findings for hyperelastic material-parameter characterization are discussed in Section 5.

TA B L E 1
First , second  invariants, and second invariant as a function of the first invariant (), for different deformation states [5,6].

NONLINEAR KINEMATICS
At first, the basic relations of nonlinear solid kinematics are introduced, which are important for the presented findings.
Immediately from the basic relations, the aforementioned kinematic deformation equivalence is calculated.The deformation of a solid is represented by the deformation gradient  and the rigid-body rotation-free part is described by the right Cauchy-Green tensor  =   .For isotropic incompressible material, the first  and second  invariants carry all relevant kinematics, here denoted with the principal stretches {  } [4]: The principal stretches or the components of the deformation gradient can be calculated under the condition of incompressibility, that is, the Jacobian of the deformation gradient is equal to one  = 1.The cartesian components for the considered deformations states uniaxial (ux), equibiaxial (bx), and plane strain (ps) result in [1]: , and where the case-dependent stretch factors  =  ∕ 0 are defined by the ratio of the deformed to the initially measured length.With these explicit deformation gradients, the first invariants () and second invariants () of Equation ( 1) are derived as a function of the stretch factor, see first columns of the Table 1.Taking the inverse function of the first invariant () and substituting the result into the second invariant, we obtain the second invariant as a function of the first invariant (), listed in the right column of the Table 1.These functions () were previously determined in [5,6] and characterize selected deformation paths within the incompressibility plane of invariants, which is shown in Figure 1.The corresponding color code used in this publication is explained in the figure caption and remains the same in the further.The gray area in the plane of invariants, enclosed by the uniaxial and equibiaxial deformation paths, represents each possible deformation state for an incompressible material [1].Subtracting three from the invariants, the origin equals the point of zero deformation.In the uniaxial case, the first invariant is dominant, and therefore, the deformation is primarily characterized by an elongation change.The equibiaxial deformation is predominantly associated with surface changes.Through the comparison of the plotted points (•) based on equidistant stretches along different deformations paths,  = [1.4,1.7, 2], two facts become evident: (i) points based on the same stretching factor  result in different invariants, see, for example, Table 2 for  = 1.4,and (ii) the distance between equidistant stretch factors along the same deformation path does not increase linearly.
The previous considerations based on nonlinear kinematics can be used to identify the necessary strain energy used during the deformation process because the strain-energy density for isotropic materials (, ) can be interpreted as strain-energy surface, see [7,8].As an example, we will consider the Mooney-Rivlin model  =  1∕2( − 3) +  2∕2( − F I G U R E 1 Incompressibility plane of invariants with selected deformation paths (): equibiaxial (red), plane strain (green), and uniaxial (blue), see Table 1.Equidistant points  = [1.4,1.7, 2] along each path are marked with (•).Points of the same first invariant  ≈ 4.07 are defined by an equivalent stretch factor  eqv and illustrated with (□).
3) that defines a plane surface.Therefore, the plane of invariants provides a powerful graphical relationship between deformation states and their influence on the strain-energy density.From Table 2, it is now straightforward to see that an equal stretch factor  leads to different strain energies, which agrees with intuition.
In the further course, the focus is on the comparison of deformation states under the condition of equal strain energy.At this point, the assumption is made that the strain-energy density  depends only on the first invariant , that is, ().The uniaxial case, for example, characterized by  ux = 1.7, is used as a reference and the intersection with the plane strain and equibiaxial deformation path are of interest, see gray line and squares (□) in Figure 1.These intersections are defined by the equivalent stretch factors  ps eqv and  bx eqv and based on the kinematic deformation equivalence of equal strain energy.If the uniaxial deformation is used as reference, they depend on the uniaxial stretch factor  eqv ( ux ).At first, the solution for  bx =  ux is presented.With the relations of Table 1, it follows and by substitution, the resulting cubic function can be solved by Cardano's formula [9] to In the same manner, the root for the plane-strain deformation state results in Both relations provide stretch factors, based on the same first invariant , cf.squares (□) in Figure 1 with  ux = 1.7,  ps eqv ≈ 1.64, and  bx eqv ≈ 1.38.This equivalence is used in [1] to precondition their tensile specimens for different deformation states in the same matter.

STRESS CONVERSION
The presented results of the nonlinear solid kinematic will be connected in this section to the stresses.As mentioned in the introduction, we are interested in converting stresses based on different deformation states into each other.Therefore, the stress-strain relations should be calculated at first.Here, isotropic hyperelastic models are considered, where the material's incompressibility is enforced using a Lagrange multiplier λ, that is,  incomp = (, ) − λ∕2( − 1).It can be determined by zero stress boundary conditions.In the general case, the first Piola-Kirchhoff stress tensor is given as With the deformation states of Equation ( 2), we obtain the principal components of the first Piola-Kirchhoff stress tensor  as listed in Table 3.The nondiagonal components are zero, cf.[4,10].Further, only the components  1 are of interest because they correspond to the stress measured during tensile tests and are suitable for verifying the conversion approach to be developed.As already mentioned, we consider first-invariant strain-energy densities (), for example, the Neo-Hookean model.Then, the stress components simplify to and  ps = 2 They can be described by a product of kinematic prefactors () and the first derivative   .Importantly, the prefactors () are independent of the material model and depend only on the nonlinear kinematic.Therefore, they are substantial for stress-stretch relations ().In Figure 2, the prefactors () for the considered deformation states are plotted.In addition, we see points of same stretch factor  = [1.4,1.7, 2] and equal first invariants  iso ≈ [3.39, 4.07, 5], illustrated with ( ) and ( ).As is known from experiments, the equibiaxial deformation state leads to the highest stress level.But by comparing points based on the same first invariant  iso , the equibiaxial state gives the smallest values and the plane strain the largest.In our case, considering only the first invariant dependence of (), these deformation points result in the same strain energy.
To compare the stress-stretch relations  ux (),  bx (), and  ps (), which result from the same strain-energy density, the prefactors () in equibiaxial and plane-strain state can be rescaled.With the results from the previous section, see Equations ( 4) and ( 5), the nested functions yield the relations Both functions are also plotted as solid lines in Figure 2. Of course, they lay on top of the original relations, but here explicit points are marked with crosses (×).These points result from the arguments  ux = [1.
Now that both the kinematic prefactors ( ux ) and the partial derivative    depend on the same first invariant , the stress functions describing the material response based on the same necessary strain energy.In terms of material characterization, where the partial derivative    is defined by material parameters to be adjusted, we receive with Equation ( 9) three coequal relations.So, we have an overdetermined system with three equations and only one unknown derivative.This statement that only one deformation state is needed to specify a first-invariant model was already introduced by [11] and is used by [12] to formulate a parameter-free material model, see Marlow model in Abaqus [13].
Here, the uniaxial deformation state is used as reference to determine the derivative with the uniaxial stress  ux to This derivative is equal for all deformations with the same first invariant .Therefore, the equibiaxial and the plane-strain stress in Equation ( 9) can be reformulated with this reference derivative to as product of the ratios of kinematic prefactors  and the uniaxial stress  ux .Plotting the ratios of prefactors  against  ux , one can see the horizontal asymptotes in Figure 3.For the plane-strain state, a direct correlation between uniaxial  ux and plane-strain stress  ps is observed.Thus, the approximation  ps conv ( ps eqv ) ≈  ux ( ux ) is valid for high stretch factors.Finally, it should be noted that the developed stress conversion approach in Equation ( 11) was derived without explicit knowledge of the strain-energy density ().

VALIDATION AND VERIFICATION
The new findings will be verified on theoretical aspects and validated on experimental data.Furthermore, we check if the kinematic deformation equivalence is really required.For this purpose, an alternative stress conversion is considered, where the reference derivative equation ( 10) is now substituted into Equation (7): F I G U R E 3 Asymptotic behavior of the kinematic prefactors  ratios.

(A) (B)
F I G U R E 4 Verification of the stress conversion approach by considering hyperelastic material models: (A) Neo-Hookean and (B) Yeoh.
The stress conversion based on  conv with deformation equivalence is denoted with (×) and Pconv is represented by (•).
The conversion method with use of the equivalent stretch factor  eqv based on Equation (11) (×) yields an exact match for both models.The alternative method according to Equation (12) produces incorrect results in the case of Yeoh model; however, correct results for the Neo-Hookean model.The reason for the different results of Equation ( 12) depends on the first partial derivative   , see Equation (10).While for the Neo-Hookean model, the derivative is constant, for the Yeoh model, it becomes a function of the first invariant .In case of a constant derivative  ux ∕ ux ( ux ) in Equation ( 12), the stress-stretch relations are the same as in Equation (7).Here,  ux is used instead of  bx or  ps , and is thus valid.But if the calculated derivative is a function of the first invariant , the numerators in Equation (12) do not correspond to an equivalent deformation.This is because the kinematic prefactors  bx ( ux ) and  ps ( ux ) are based on different first invariants  compared to the derivative.Therefore, the deformation equivalence is required to rescale the kinematic prefactors  bx and  ps in Equation (11), in the general case.

(A) (B)
F I G U R E 5 Validation of the stress conversion approach by considering two experimental data sets (A) German Institute for rubber technology and (B) Axel Products Inc. [14].The (×) corresponds to stress conversion  conv with deformation equivalence.
Finally, the stress conversion approach is verified using experimental data with carbon black filled rubbers.At the Deutsches Institut für Kautschuktechnologie e. V. (German Institute for rubber technology), a data set was determined with a strain rate of 1.7 % ∕s without preconditioned specimens.The second data set was characterized by Axel Products Inc. and is found in the literature [14].Here, three samples are tested in each deformation state up to an elongation of 25% with a strain rate of 0.01 % ∕s.A multihysteresis test setup was chosen to obtain the considered stabilized loading paths.The remaining strain offset was removed in the postprocessing, see [14].Both data sets differ in the applied strain rate and the specimens precondition and therefore also in more or less viscoelastic and damage effects.The measured stress-stretch relations are plotted with solid lines in Figure 5, whereby the leak in the equibiaxial data in Figure 5(a) is caused by initial load reconstruction based on a multihysteresis test.Error bars based on a 95 % confidence interval (CI) are presented for the uniaxial and plane-strain states.
The comparison of the converted stresses (×) with the real data shows a good agreement in both data sets.Due to the different experimental setups, the stress transformation approach appears to be independent of additional effects such as viscoelasticity or damage.For further statements, however, the hyperelastic and viscoelastic effects should be separated, see [15].

CONCLUSION
In this work, an approach to stress conversion between different deformation states was derived.With the assumption of first-invariant strain-energy densities (), we were able to find a kinematic deformation equivalence, which result in the same strain energy for the uniaxial, equibiaxial, and plane-strain deformation states.The stress conversion approach was formulated by linking this deformation equivalence to the basic equations of hyperelasticity.The following verification and validation confirmed the validity of the procedure.
Because we are now able to reformulate the further stress-strain relations, additional tensile tests based on further deformation states than the uniaxial case can be omitted for first-invariant material models.However, a multiaxial test is still required to verify the -dependence of an unknown material.Furthermore, once -dependence is demonstrated for a material, multiaxial test systems can be verified.The converted stresses should then agree with the multiaxial test data.
For a more general validity, research is ongoing to extend the presented approach to a second invariant dependence.

A C K N O W L E D G M E N T S
The authors would like to thank Abhiram Sarmukaddam and Alexander Ricker for their helpful assistance in conducting the material testing at Deutsches Institut für Kautschuktechnologie e. V. (German Institute for rubber technology).
Open access funding enabled and organized by Projekt DEAL.
Example values for invariant pairs ( − 3,  − 3) based on different deformations states with a stretch factor  = 1.4.
Principal components of the first Piola-Kirchhoff stress tensor  for isotropic incompressible hyperelastic models.