When compressed, hyperelastic materials such as non-woven textiles produce monotonous stress-strain curves, i.e., in which no linear part can be identified. In such conditions, is it meaningful to determine a compressive modulus, since the slope of the compression curve continuoulsy increases with strain ? If no modulus can be determined, how can we characterise more or less flexible hyperelastic materials ? Which parameters can we derive from such curves in order to compare hyperelastic materials with each other ?
Hello!
For example, with a deformation of 10%, the first material can be more flexible, and with a deformation of 20% the second material can be more flexible.
Just need to compare the level of strain and the level of stress various materials.
non-linear elastic materials can be assessed foe elastic modulus by initial tangent modulus or secant modulus or tangent modulus - last two at a given strain.
I propose two methods to compare the modulus of such materials :
1- Average modulus at a certain deformation, which is obtained from dividing stress(strength) by deformation at an optional deformation.
2-Initial modulus which is the slope of the stress-strain curve at deformation=0%.
Both methods are relative and if the results are different at different deformations, it is better to compare the modulus in separate zones of deformation.
There are different constitutive equations that characterize hyper elastic behavior. In Simples cases a so called Neo-Hooke model that depends on the first and third invariant of the strains can be used. It has two parameters that can be used to match the measured response curves. If this is not successful then the more general class of Ogden materials can be applied, see pages 13-18 in the appended text.
@Peter : is there any simple equation able to fit the strain-stress curve ? I am not specialist at all and I cannot see anything "simple" in the document you kindly provided.
Alain
The compressional hyperelastic behavior in 3D can be described by the following equation:
Tau=2 * (mu/alpha * (Balpha/2 - J-alpha*beta)),
with 3 parameters of hyperelasticity: mu, alpha, and beta
where Tau is the Kirchhoff tensor, J is the Jacobian, B is the square of the deformation gradient tensor F.
mu and alpha control the isotropic stiffness (mu at the beginning of the stress-strain curve, alpha - afterwards). beta is responsible for the compressibility.
If 1 is the compressional direction, then:
Tau11=2 * mu/alpha * (lambda1alpha - (lambda1lambda2lambda3)-alpha*beta)
if the deformations lambda2, lambda3 in in-plane directions also take place.
Best regards
Thank you All for your comments.
However, I was looking for a simple law such as stress = f(strain) with a few adjustable parameters, that I might fit to my experimental data, and from which the parameters would bring some useful information on the material ...
Does such simple descriptive law exists ?
Alain
Dear Alain,
first of all, the constitutive equations are three-dimensional and must be reduced to the boundary-value problem of a 1D-case. According to Peter Wriggers proposal this can be done for particular strain.energy functions. If you use the assumption of incompressibility, it leads to more or less simple expressions. For the case of a compressible material you will obtain two coupled equations with two unknowns. In this case it is slightly more complicated to identify the material parameters. If you are interested in the incompressible case, it is documented in
S.H.: Parameter estimation of hyperelasticity relations of generalized polynomial-type with constraint conditions, International Journal of Solids and Structures 38 (44-45), 2001, 7999-8018
and the literature cited therein.
Stefan
Thank you Stefan,
Just one more question for my information and curiosity: how can a hyperelastic material be incompressible ? What does it mean ? Our materials (felts and fibrous mats) can be compressed without effort (hence an apparently zero modulus), and they completely recover when removing the compression stress, so how would they be incompressible ?
It seems here that compressible / incompressible has a different meaning than common sense ...
Thanking you in advance for your explanation,
Alain
Dear Alain,
if a material is incompressible, it does not imply that there is no deformation. It means only, that the volume remains constant under deformation. For example, a fluid is commonly assumed to be incompressible. Frequently, rubber is also modeled incompressible. However, if you have a porous material incompressibility would be a very rough assumption. You have to measure the lateral strains (deformation) in your compression experiment as well. If the material behaves homogeneous (constant strains within a certain region), you can relate axial and lateral deformations to find out whether the assumption of incompressibility is justified.
If you remove your load and obtain the same stress-strain curve as in the loading case, you can roughly assume an elastic material (which has nothing to do with compressibility).
Kind wishes
Stefan
For fibrous materials behavior we observe the hysteresis and permanent deformation in loading/unloading present due to the irreversible redistribution of fibers at micro-scale.
Best regards
For many nonlinear elastic materials, the elastic responses are generally functions of the deformation. In particular, for simple tension or compression, a nonlinear elastic modulus can be defined to measure changes of the stresses with changes of the strains. See for example the nonlinear elastic modulus E given by eq. (2.9), with mu0 defined for various popular hyperelastic models (neo-Hookean, Mooney-Rivlin, Fung, Gent, Ogden) in Sec. 2.3, of the attached paper. For a more general discussion, see also the section on "Stress increments and elastic moduli" in the monograph "Non-linear elastic deformations" by R.W. Ogden.
Hello Alain Celzard,
You can unravel the theory of deforming a thin membrane. It discusses point shifts at constant deformation. Part of this theory is affected in our articles.
Article Research highligths of sheet metal testing by hydraulic bulging
Article SOME FEATURES OF THE SHEET METALS TESTING BY HYDRAULIC BULGING
Research Тhesis: SHEET MATERIALS TESTING USING BIAXIAL HYDRAULIC BULGE TEST
Dear Alain,
General Theory of Continuum Media Thermodynamics show the possibility to treat hyperelastic materials using a Langrangian Approach. So using different constitutive hypelastic equations as the neo-Hookean, Mooney-Rivlin and others you can identify all parameters (compression modulus , etc) using a non-linear identification method. You can see same books of Continuum Media Theory (especially last edition book of COIRIER) and especially description and documents of practic hyperelastic theory from ASTER finite element code (free code of EDF - France).
Adinel GAVRUS
Dear Colleagues,
I think, it is necessary to lubricate the interface sample and platens during the compression test in order to obtain a quasi-homogeneous strains through the sample (see Jerabeck et al. Polymer testing in the case of PTFE....), so a cylinder is transformed into "cylinder". If no lubricate is used then the bulging occurs during compression, in my opinion, the measured deformation depends on the marterial point (so, the deformation is heterogenous).
H. Bechir.
Thank you all for good discussions.First of all, hyperelasticity is the property of materials in their elastic deformations. It means that energy given to material is completly returned while eliminating external load, regardless of its type and form of impact. Hyperelasticity is the property of material and should be invariant in relation to types of external impact. If material is hyperelastic so, it is possible to study the tasks of the stressed state in the defined external impacts. Flexibility is the property that relates to construction not to materials. Therefore, one cannot mix it with elasticity of materials. It is necesary to correctly distinguish linear part and determine the required moduli from your curves. Linear part (if it is available) is the indicator of linear deformations. It is necessary to determine moduli of elascticity using this part of your graphs. Nonlinear part is the indicator of nonlinear deformations. They are not related to elasticity moduli of materials.
Comprhensive answer has not given into this question in discussions. The problem is fundamental character. Therefore, an appropriate answer should be complete. It is mainly necessary for young researchers. So, I additionally answer into this question:
The main property is the reversibility of all the processes occurring in it after removal of loads causing these processes for an elastic body.
A hyperelastic body is usually called an elastic body, for which the expression of potential energy is specified and the correlations between the components of the stress and strain tensors for compressible and incompressible elastic bodies with finite and small deformations are constructed on its basis.
Thermodynamic potentials— the internal energy as a function of the components of the Green deformation tensor and entropy, and the free energy as a function of the components of the Green deformation tensor and temperature are specified to describe hyperelastic bodies.
The condition of incompressibility is additionally satisfied for incompressible media, which gives the correlation between the variations of the components of the Green deformation tensor.
The correlations of the components of the stress and temperature tensor on the components of the strain tensor, entropy and scalar function related with hydrostatic pressure are obtained performing this scheme in the case of specifying the expression for the internal energy. The correlations of the components of the stress and entropy tensor on the components of the strain tensor, temperature, and the scalar function are related to hydrostatic pressure specifying the expression for the free energy.
It is possible to introduce a single function of the specific potential deformation energy called the elastic potential for adiabatic and thermal processes instead of two thermodynamic potentials.
It should be emphasized that the correlations obtained within the theory of finite deformations in the case of small deformations are simplified in a known way.This scheme is suitable for the elementary volume and the basic values refer to the unit volume of the undeformed state.
Best wishes..
İ congratulate you with new and good book. I think this book would be useful for young scientists and researchers. I wish you following succesess.
If you have a stress-strain curve (Lagrangian especially or Eulerian if necessary), obtained from any experimental test, in particular compression one, you can compute, for a defined maximal strain, the ratio between the total elastic energy (integral of stress_strain) and the equivalent elastic energy obtained from the equivalent total elastic energy computed from integral of linear stress_strain curve using the slope equal to the initial apparent elastic modulus E0 (slope of real stress_strain curve at a strain equal to 0). This ratio can be use to conclude if a hyperelastic material is more flexible or more rigid that a chosen reference one (caoutchouc). The most easy is to compute the ratio between the total elastic energy of the real material (integral of stress_strain curve) with respect to the reference material one (caoutchouc). If the ratio is more great the the rerence material one, the studied material is more rigid that this one (caoutchouc) or more flexible if the ratio is smallest.... Adinel GAVRUS (INSA Rennes, France)
Do you have any experimental data such as force and deformation? If so, then you can calculate the stretch ratio. Using Mooney-Rivillin hyperplastic model or Neo-Hookean model you can calculate the material constant from which you can easily calculate the E or G value.
Using Non-Linear Regression on Force- Displacement curves using a defined hyperelastic models (Hookean, Monney-Rivllin, etc) to make parameter's identification which characterises the constitutive model.
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