With symmetries measured by the Lie group and curvatures revealed by differential geometry, the continuum stored energy function possesses a translational deformation component, a rotational deformation component, and an ellipsoidal volumetric deformation component. The function, originally developed for elastomeric polymers, has been extended to model brittle and ductile polymers. The function fits uniaxial tension testing data for brittle, ductile, and elastomeric polymers, and elucidates deformation mechanisms. A clear distinction in damage modes between brittle and ductile deformations has been captured. The von Mises equivalent stress has been evaluated by the function and the newly discovered break-even stretch. Common practices of constitutive modeling, relevant features of existing models and testing methods, and a new perspective on the finite elasticity-plasticity theory have also been offered.
Constitutive modeling for finite deformations of polymeric materials requires accurate theoretical predictions combined with experimental characterizations. Experimental tests cannot characterize materials in all deformation modes although experimental tests are more complete than theoretical models. As a result, a theoretical model is fitted with experimental data tested in certain deformation modes and the fitted model predicts deformations in untested modes. Thus, the theoretical development of constitutive models and their numerical implementations into finite element methods are crucial in designs and analyses of polymer components, which have been heavily applied in the aerospace, automotive, consumer electronics, medical device, and space exploration industries.
For elastic-plastic constitutive models, great progress has been made in continuum mechanics. Earlier achievements in kinematics, conservation principles, and constitutive relations were archived in the classic treatise on continuum mechanics by Truesdell and Noll (1965) [
Incremental strain as well as total strain theories of plasticity, featuring assumptions, models, and computer implementations, have been comprehen- sively reviewed by Armen (1979) [
Accurate constitutive relations and further computer implementations are crucial to maintaining the required numerical properties of accuracy, efficiency, and stability. Among them, accuracy is one of the most important numerical properties. The five key elements of existing elastic-plastic constitutive relations are stress-strain relations, decompositions of strains, yield criteria, hardening laws, and flow rules. The current theories of plasticity can only maintain the numerical stability in terms of normality of the incremental plastic strain vector and convexity of initial and subsequent yield surfaces but cannot accurately predict plastic deformations in general modes, as noticed by Michno and Findley (1976) [
The continuum stored energy (CSE) function, originally developed for modeling elastomeric polymers by Zhao (2016) [
For isothermal processes, the general CSE function for isotropic materials at finite deformations, Ψ , has been postulated to be balanced with its stress work done as1
Ψ = S : C 2 (1)
where the second Piola-Kirchhoff stress tensor, S , reads
S = 2 [ ( ∂ Ψ ∂ I 1 + I 1 ∂ Ψ ∂ I 2 ) I − ∂ Ψ ∂ I 2 C + I 3 ∂ Ψ ∂ I 3 C − 1 ] (2)
the right Cauchy-Green tensor, C , is given by
C = F T F (3)
and F is the deformation gradient tensor.
The three invariants of right Cauchy-Green tensor, I 1 , I 2 , and I 3 , are related by
I 1 = tr C = λ 1 2 + λ 2 2 + λ 3 2 (4)
I 2 = 1 2 [ ( tr C ) 2 − tr C 2 ] = I 3 tr C − 1 = λ 1 2 λ 2 2 + λ 2 2 λ 3 2 + λ 3 2 λ 1 2 (5)
I 3 = det C = λ 1 2 λ 2 2 λ 3 2 (6)
the trace of the tensor C is denoted by tr C , the determinant of the tensor C is denoted by det C , and λ 1 , λ 2 , and λ 3 are principal stretches in three mutually orthogonal directions. The stretch vector λ a 0 in the direction of the unit vector a 0 is defined as
λ a 0 = F a 0 (7)
with the length λ = | λ a 0 | , which has been documented by Holzapfel (2000) [
1The general CSE function, Ψ , is covariant to Ψ E = S : E under the transformation of E = ( C − I ) / 2 , which has been emphasized by Zhao (2016) [
Substituting (2) into (1), simplifying, and rearranging yields the following partial differential equation
Ψ = I 1 ∂ Ψ ∂ I 1 + 2 I 2 ∂ Ψ ∂ I 2 + 3 I 3 ∂ Ψ ∂ I 3 (8)
Based on Lie group methods, the characteristic system for the partial differential Equation (8) is
d I 1 I 1 = d I 2 2 I 2 = d I 3 3 I 3 = d Ψ Ψ (9)
and taking its three first-integrals, ψ 1 = I 2 / I 1 2 , ψ 2 = I 3 / I 1 3 , and ψ 3 = Ψ / I 1 , the general solution has been obtained and written as
Ψ = I 1 [ f ( I 2 / I 1 2 ) + g ( I 3 / I 1 3 ) ] + c 0 (10)
where f and g are two arbitrary functions, c 0 is an integration constant, and the general solution defines a group of CSE functions.
The general solution (10) has two arbitrary functions to be determined for practical applications. Translational, rotational, and ellipsoidal deformations are indispensable components for isotropic polymeric materials. Thus, the curvatures of the three types of deformations have been used to select the two arbitrary functions. By doing so, the particular CSE function turns out to be
Ψ p ( I 1 , I 2 , I 3 ) = c 0 + c 1 I 1 + c 2 I 2 + c 3 I 1 4 I 3 (11)
where the three coefficients, c 1 , c 2 , and c 3 , are unknown constitutive constants to be determined by experimental tests.
For the initially undeformed referential configuration, we have the normalized stretches of λ 1 = λ 2 = λ 3 = 1 , resulting in the three constant invariants of I 1 = I 2 = 3 and I 3 = 1 from (4), (5), and (6). The CSE function at the unde- formed mode is usually assumed to be zero
Ψ p ( 3 , 3 , 1 ) = 0 (12)
Substituting the c 0 determined by (12) into (11) and collecting terms yields the following three-component CSE function
Ψ p ( I 1 , I 2 , I 3 ) = c 1 ( I 1 − 3 ) + c 2 ( I 2 − 3 ) + c 3 ( I 1 4 I 3 − 81 ) (13)
For curve fittings with the CSE function (11), the engineering stress and the true stress as a function of principal stretches in uniaxial tension have been derived, respectively. With the constraint of incompressibility, the engineering stress in uniaxial tension tests, P u , as a function of principal stretches reads
P u = [ 2 c 1 + 1 2 λ 3 + 1 c 2 + 8 ( λ 2 + 2 λ − 1 ) 3 c 3 ] ( λ − λ − 2 ) (14)
and the true stress in uniaxial tension tests, σ u = P u λ , as a function of principal stretches is
σ u = [ 2 c 1 + 1 2 λ 3 + 1 c 2 + 8 ( λ 2 + 2 λ − 1 ) 3 c 3 ] ( λ 2 − λ − 1 ) (15)
The true stress in equibiaxial tension tests, σ b , as a function of principal stretches is
σ b = [ 2 c 1 + 1 1 + 2 λ − 6 c 2 + 8 ( 2 λ 2 + λ − 4 ) 3 c 3 ] ( λ 2 − λ − 4 ) (16)
The true stress in pure shear tests, σ s , as a function of principal stretches is
σ s = [ 2 c 1 + 1 λ 2 + λ − 2 + 1 c 2 + 8 ( λ 2 + λ − 2 + 1 ) 3 c 3 ] ( λ 2 − λ − 2 ) (17)
For finite elastic-plastic deformations of semi-crystalline polymers, there exists a break-even stretch, λ b , between translational and rotational deforma- tion curves in uniaxial tension tests. Equating the first term and the second term from either (14) or (15) and solving gives
λ b = 1 2 [ ( c 2 c 1 ) 2 − 4 ] 1 3 (18)
The break-even stretch will be used to study the von Mises equivalent stress.
The deformation mechanisms in polymers are usually classified as brittle, ductile (necked and unnecked), and elastomeric, as documented by Meyers and Chawla (2009) [
General-purpose polystyrene (GPPS), as a brittle polymeric material, deforms mainly elastic and breaks at about 2% strain, which can be found in the book by Smith (1993) [
The two ductile polymeric materials selected for uniaxial tension tests are low-density polyethylene (LDPE) and isotactic polypropylene (iPP). Both LDPE and iPP are semicrystalline polymers, containing both amorphous and crystalline phases. Under uniaxial tensions with constant elongation rates and temperatures, semicrystalline polymers are characterized as elastic-plastic materials. The true stress Equation (15) for uniaxial tension tests will be used to model ductile polymeric materials.
The uniaxial tension test of LDPE conducted at the elongation speed of 15 mm/ min by Nitta and Yamana (2012) [
The uniaxial tension test of iPP conducted at the elongation speed of 5 mm/ min by Nitta and Yamana (2012) [
Rubber exhibits finite elastic deformations under tensile loading and fully recovered deformations after unloading.
The benchmark experimental data of vulcanized rubber containing 8% sulfur was accomplished by Treloar (1944) [
The general solution (10) of the CSE partial differential Equation (8) establishes the generic CSE function for isotropic materials under isothermal processes. Based on the requirements of translational, rotational, and ellipsoidal deformations, the general solution boils down to a particular three-component solution, in which mechanical behavior of materials is concisely described by the three constants: c 1 , c 2 , and c 3 . In the three-component CSE function (13) for isotropic materials, the first component, c 1 ( I 1 − 3 ) , represents the work done of normal stress and translational deformation. The second component, c 2 ( I 2 − 3 ) , describes the work done of shear stress and rotational deformation. The third component, c 3 ( I 1 4 / I 3 − 81 ) , captures the work done of volumetric stress and ellipsoidal deformation.
Physical relevancies of the CSE function are further demonstrated through uniaxial tension tests with the fitted constants for selected polymeric materials with different deformation mechanisms. In addition to the representative materials, the curve-fitting results of PMMA (PLEXIGLAS), HDPE by Nitta and Yamana (2012) [
For the GPPS row in
2A polymer fails by either shearing or crazing, as pointed out by Kausch (1987) [
Damage modes for polymers2. have been revealed through the CSE function. The CSE function predicts that the fractures of brittle material GPPS are due to both shearing and ellipsoidal volumetric damaging. The second term in the continuum constitutive model corresponds to a shear stress and a rotational
deformation. Thus, the negative second term describes the damage in shearing mode. The third term in the continuum constitutive model corresponds to a volumetric stress and an ellipsoidal deformation. Thus, the negative third term captures the damage in the ellipsoidal volumetric opening mode. This mode will be further described in ductile polymer cases.
For the LDPE and iPP rows in
The positive values of c 1 , c 2 , and c 3 for rubber under uniaxial tension tests predict no damage in the combination of translational, rotational, and ellipsoidal volumetric deformations within the test range.
The c 1 I 1 term was used in modeling elastomeric polymers. Stored energy functions with only I 1 terms, however, will fail to model simple torsion experiments for isotropic elastomeric polymers demonstrated by Horgan and Saccomandi (1999) [
The c 3 I 1 4 / I 3 term captures the ellipsoidal volumetric strengthening for elastomeric polymers at finitely large stretches.
3The three stages of ductile fractures are void nucleation, void growth, and void coalescence, as summarized by Garrison and Moody (1987) [
The key to the yield criterion, hardening law, and flow rule formulations is the von Mises equivalent stress. Thus, the von Mises equivalent stress and strain play a crucial role in existing theories of plasticity documented by Khan and Huang (1995) [
S ¯ = 3 2 S ′ : S ′ = 3 2 S : S − 1 2 ( tr S ) 2 (19)
and
E ¯ = 2 3 [ ln ( λ 1 λ 2 ) ] 2 + [ ln ( λ 2 λ 3 ) ] 2 + [ ln ( λ 3 λ 1 ) ] 2 (20)
where S ′ is the deviatoric stress tensor, S ¯ is the von Mises equivalent stress, and E ¯ is the equivalent logarithmic strain in terms of principal stretches.
Stress-based formulations, however, possess some significant shortcomings. Substituting (2) into (19)2, applying the Cayley-Hamilton equation, collecting terms, and simplifying yields the von Mises equivalent stress in terms of the CSE function
S ¯ = 2 ( I 1 2 − 3 I 2 ) ( ∂ Ψ ∂ I 2 ) 2 + ( I 2 2 − 3 I 1 I 3 ) ( ∂ Ψ ∂ I 3 ) 2 + ( I 1 I 2 − 9 I 3 ) ∂ Ψ ∂ I 2 ∂ Ψ ∂ I 3 (21)
where no ∂ Ψ / ∂ I 1 related terms are noticed.
The Kirchhoff stress tensor, τ , can be converted from the second Piola-Kirchhoff stress tensor by the push-forward operation by F
τ = I 3 σ = F S F T (22)
where σ is the Cauchy stress tensor or the true stress tensor.
With the CSE function and I 3 = 1 , the von Mises equivalent stress (21) in terms of invariants boils down to
S ¯ p = c 2 I 1 2 / I 2 − 3 (23)
The c 2 term in (23) indicates that shear stresses drive elastic-plastic deformations. Clearly, the contribution of the c 1 and c 3 terms in the CSE function is excluded from the von Mises equivalent stress equation since it was formulated on the assumption that plastic deformations depend only on shear stress instead of normal stress and volumetric stress. Many studies in the past and present show that the von Mises equivalent stress does not produce very good results for isotropic as well as anisotropic ductile metallic materials by Michno and Findley (1976) [
In order to better understand finite elastic-plastic deformations of polymers, c 1 , c 2 , and c 3 terms in the continuum constitutive model are individually depicted for both LDPE and iPP ductile polymers. The true stress-stretch component curves for both LDPE and iPP are shown in
4Deformations of ductile polymers are the chain orientation and the volume damage, mentioned by Ponçot, Addiego, and Dahoun (2013) [
All three terms of the continuum constitutive model contribute to elastic- plastic deformations.4 The first term and the second term are the major contributors to elastic-plastic deformations, meaning that both the normal stress and the shear stress contribute to elastic-plastic deformations. As the stretch increases, both normal stress and shear stress increase. Between the curves of the first term and the second term, there exists a break-even stretch, λ b , defined in (18). When principal stretches are much smaller than the break-even stretch, the rotational deformation dominates elastic-plastic deformations and shear stress contributes more than normal stress. At the break-even stretch, shear stress
contributes the same as normal stress. When principal stretches are much greater than the break-even stretch, the translational deformation dominates further elastic-plastic deformations and normal stress contributes more than shear stress. The third term only contributes minor volumetric damages at very large stretches. The break-even stretches for the selected ductile polymers have been evaluated and determined as λ b = 6.34374 for LDPE and λ b = 25.24009 for iPP, respectively. As stretches fall within 0 and 1, deformations are compressive. In compressive deformations, shear stress indeed dominates plastic deformations while normal stress contributes little as predicted and depicted in both
The equivalently converted stress-strain curves in different deformation modes should coincide with its uniaxial tension stress-strain curve for both elastic and elastic-plastic deformations. The equivalent stress (21), along with (22), and the equivalent logarithmic strain (20) for Treloar’s experimental data in uniaxial tension, equibiaxial tension, and pure shear modes have been calculated and depicted in
Unlike the von Mises equivalent stress or the equivalent strain, the CSE function captures the finite elastic-plastic deformations of ductile polymers in the following aspects:
1) Different combinations of three coefficients for the selected ductile polymers given in
2) There exist tensile break-even stretches for finite deformations of the selected ductile polymers defined in (18) and shown in
3) The values of c 3 in
4) The combination of three invariants of stretches in the CSE function (11) possesses three different deformation components, generating physically relevant deformations.
5) The CSE function is a physically relevant and mathematically covariant model established from continuum mechanics, predicting deformations in other untested modes.
Thus, a new algorithm based on the CSE function for the finite elasticity- plasticity theory is recommended for future research and development.
The most commonly used experimental tests for material characterizations are uniaxial tension, pure shear, equibiaxial tension, uniaxial compressions, confined compression, and torsion tests, as reviewed by Charlton, Yang, and Teh (1994) [
Common practices to characterize and model elastomeric polymers are to take the uniaxial tension, pure shear, and equibiaxial tension tests, to fit several theoretical models with the three test results, and to select the best possible theoretical model based on accuracy, stability, and efficiency. Among the three tests, the pure shear test can produce the same accuracy as the uniaxial tension test, but the deformation range is far less than that of the uniaxial tension. The equibiaxial tension fails to match both the accuracy and the deformation range of the uniaxial tension. Thus, the uniaxial tension test is the best method available to characterize the mechanical behavior of materials. When theoretical constitutive models are fitted by the test in one mode, predictions of other deformation modes will indicate the quality of theoretical models, as noticed by Steinmann, Hossain, and Possart (2012) [
To characterize and model finite deformations of ductile polymers, uniaxial tension and compression tests are usually used. Intrinsic material nonlinearities, however, could be coupled with boundary and geometric nonlinearities during experimental tests. In order to capture intrinsic material nonlinearities, symmetries should also be applied in experimental designs and tests. For necked specimen in uniaxial tension tests, their deformations are no longer homo- geneous and traditional extensometers are no longer accurate. Video-based extensometers by G’sell and Jonas (1979) [
In the CSE constitutive modeling for isotropic polymeric materials, both experimental characterizations and theoretical predictions are indispensible. Experimental tests are used to determine the three unknown constitutive constants in the CSE function for a specific material. The CSE function as a theoretical model is applied to predict the deformations of untested deformation modes since exclusive experimental tests cannot cover all possible deformation modes. For isotropic polymeric materials with different deformation me- chanisms, applications of the CSE function and uniaxial tension tests minimize errors and tests in traditional constitutive modeling processes.
The CSE function, originally developed for modeling elastomeric polymers, has been extended in its application to brittle and ductile polymers. The CSE function fits uniaxial tension data for GPPS and PMMA as brittle polymers, LDPE, iPP, and HDPE as ductile polymer materials both with and without necking, and vulcanized rubber containing 8% sulfur and Entec Enflex S4035A TPE as elastomeric polymers remarkably well.
The CSE function as a physically relevant and mathematically covariant model elucidates the three major deformation mechanisms of polymers. For brittle polymer materials, deformations are mainly elastic, then slightly elastic-plastic deformations before fracture, and the failure modes are both shearing and ellipsoidal volumetric opening. For ductile polymeric materials, deformations begin with slightly elastic, then mainly elastic-plastic, and ellipsoidal volume damages at relatively large stretches. For elastomeric polymeric materials, deformations are rather homogeneous and fully elastic with no damages within the testing range.
The difference between brittle and ductile elastic-plastic deformations is that brittle polymers take little shearing deformation while ductile polymers endure both finite translational deformation by normal stress and rotational deformation by shear stress.
The von Mises equivalent stress has been studied in terms of the CSE function. In large enough tensile deformations, all three terms in the continuum constitutive model contribute to elastic-plastic deformations and a break-even stretch exists between the normal stress component curve and the shear stress component curve. The von Mises equivalent stress is only approximately accurate within a narrow stretching range.
Common practices in constitutive modeling of polymeric materials have been briefly reviewed, the advantages and disadvantages of existing models and testing methods have been concisely discussed, and the CSE function and uniaxial tension tests have been selected for the constitutive modeling of polymeric materials. A new algorithm based on the CSE function for the finite elasticity-plasticity theory of ductile polymers has been evident.
The author is immensely grateful to Ampere A. Tseng and Tei-Chen Chen for their encouragement. He is also very thankful to Youyi Chu for useful discussions in the field of materials science.
He would also like to show his gratitude to Jianming and Jiesi Zhao for their help in many ways.
Zhao, F.Z. (2017) The Continuum Stored Energy for Constitutive Modeling Finite Deformations of Polymeric Materials. Advances in Pure Mathematics, 7, 597-613. https://doi.org/10.4236/apm.2017.710036