Biomed Tech 2014; 59(5): 439–446
Alireza Karimi, Mahdi Navidbakhsh*, Mansour Alizadeh and Reza Razaghi
A comparative study on the elastic modulus of polyvinyl alcohol sponge using different stressstrain definitions Abstract: There have been different stress-strain definitions to measure the elastic modulus of spongy materials, especially polyvinyl alcohol (PVA) sponge. However, there is no agreement as to which stress-strain definition should be implemented. This study was aimed to show how different results are given by the various definitions of stress-strain used, and to recommend a specific definition when testing spongy materials. A fabricated PVA sponge was subjected to a series of tensile tests in order to measure its mechanical properties. Three stress definitions (second Piola-Kichhoff stress, engineering stress, and true stress) and four strain definitions (AlmansiHamel strain, Green-St. Venant strain, engineering strain, and true strain) were used to determine the elastic modulus. The results revealed that the Almansi-Hamel strain definition exhibited the highest non-linear stressstrain relation and, as a result, may overestimate the elastic modulus at different stress definitions (second Piola-Kichhoff stress, engineering stress, and true stress). The Green-St. Venant strain definition failed to address the non-linear stress-strain relation using different definitions of stress and invoked an underestimation of the elastic modulus values. Engineering stress and strain definitions were only valid for small strains and displacements, which make them impractical when analyzing spongy materials. The results showed that the effect of varying the stress definition on the maximum stress measurements was significant but not when calculating the elastic modulus. It is important to consider which stress-strain definition is employed when characterizing *Corresponding author: Mahdi Navidbakhsh, School of Mechanical Engineering, Iran University of Science and Technology, Tehran 16846, Iran, Phone: +00982177209027, Fax: +00982173021585, E-mail: [email protected]; and Tissue Engineering and Biological Systems Research Laboratory, School of Mechanical Engineering, Iran University of Science and Technology, Tehran 16846, Iran Alireza Karimi and Reza Razaghi: Tissue Engineering and Biological Systems Research Laboratory, School of Mechanical Engineering, Iran University of Science and Technology, Tehran 16846, Iran; and School of Mechanical Engineering, Iran University of Science and Technology, Tehran 16846, Iran Mansour Alizadeh: School of Mechanical Engineering, Iran University of Science and Technology, Tehran 16846, Iran
the mechanical properties of spongy materials. Although the true stress-true strain definition exhibits a non-linear relation, we favor it in spongy materials mechanics as it gives more accurate measurements of the material’s response using the instantaneous values. Keywords: Almansi-Hamel strain; polyvinyl alcohol sponge; second Piola-Kirchhoff stress; stress-strain definition; tensile testing. DOI 10.1515/bmt-2013-0110 Received October 16, 2013; accepted March 7, 2014; online first March 29, 2014
Introduction Polyvinyl alcohol (PVA) sponges are currently in widespread use for the removal and management of diffuse fluids/blood at the surgical site [18]. They are also contemplated as the most attractive biomedical polymers owing to a combination of qualities, such as biocompatibility [2, 20, 24, 27], high hydrophilicity [19, 23, 25], excellent mechanical strength and flexibility [17, 19, 20, 25–27], thermal stability and absence of toxicity [21], availability, and relative cheapness [22]. However, the application of this versatile material has been limited to ophthalmic, plastic, and hand surgeries as a biocompatible biodegradable material. Knowledge of the mechanical properties of the PVA sponge may pave the way to find a suitable application in tissue engineering as scaffolding material. Several studies have been conducted to measure the mechanical properties of PVA sponge using biaxial puncture and uniaxial tensile test [12]. They showed beneficial results in terms of stress failure compared with two commercially available sponges. Karimi et al. [11] characterized the mechanical properties of a fabricated PVA sponge (P-sponge) using a uniaxial tensile test for tissue engineering applications. They showed the Young’s modulus and maximum stress of 40 and 9.79 MPa for the PVA sponge, respectively. The Young’s modulus of PVA sponge was measured at different
Brought to you by | University of Manitoba Authenticated Download Date | 6/9/15 6:08 PM
440 A. Karimi et al.: Stress-strain definitions for elastic modulus measurement strain rates to investigate the effects of strain rate on the mechanical properties of spongy materials [6]. The results showed a Young’s modulus of 4.28, 208.33, and 187.51 MPa for the strain rate of 1, 20, and 100 mm/min, respectively. PVA sponges also showed different mechanical properties under longitudinal and circumferential loading directions. The Young’s modulus of the PVA sponge in the longitudinal and circumferential directions was 38.91 and 33.34 MPa, respectively. In addition, the maximum stress in the longitudinal direction was 17.90% greater than that in the circumferential direction [4]. The mechanical behavior of PVA sponge has been shown to be similar to rubber-like materials, such as time-dependent viscoelastic behavior, which can be formulated by a visco-hyper elastic approach under low-strain uniaxial loading [5, 7, 9, 12]. Almost all the measurements on the mechanical properties of spongy materials, especially PVA sponge, have used true stress-true strain or engineering stress-engineering strain definitions. However, there is no agreement about which stress-strain definition should be adopted to determine the mechanical behavior of spongy materials in a tensile testing machine. The stress-strain definition may substantially affect the measured mechanical properties, including elastic modulus, maximum stress, and strain. The main objective of this study is to show the results given by different definitions of stress-strain and to rationalize our recommendation for the use of a specific stress-strain definition to test spongy materials. Three stress definitions (second Piola-Kirchhoff stress, engineering stress, and true stress) and four strain definitions (Almansi-Hamel strain, Green-St. Venant strain, engineering strain, and true strain) are used in this study.
poured into Petri dishes and allowed to stand at room temperature (25–30°C) until cross-linking was completed (48 h).
Materials and methods
Engineering strain
Materials and specimen preparation The preparation of the PVA sponge has been thoroughly described in our previous studies [4, 9]. Briefly, to prepare the PVA aqueous solution, 2 g of PVA (molecular weight = 40,000; Sigma-Aldrich, St. Louis, MO, USA) was dissolved in 100 ml of distilled water at 50°C under stirring at 400 rpm for 6 h. The polymer solution was then cast into cylindrical molds and freeze-dried in order to obtain a PVA spongy matrix. To improve its stability in water, the above sponge was cross-linked by exposure to the vapors of a glutaraldehyde aqueous solution (25%) at 37°C for 24 h. After being rinsed with distilled water, the sponge was freeze-dried again. The final solution was
Axial measurements The initial dimensions of all specimens were measured precisely. The tensile test was performed using a uniaxial tensile test apparatus adapted for testing biological specimens used in our previous studies [8, 10, 14, 15]. All tests were performed at 25°C, and each sample was tested only once. A low strain rate of 5 mm/min, which is typical for surgical procedures and gives more insight into tissue behavior, was employed by the action of an axial servo motor [13]. Adhesive sandpaper of fine grit was placed on the surface of the jaws of the pneumatic grips to prevent slippage. Thereafter, the samples were subjected to a continuous tensile force until failure.
Strain definitions Engineering or conventional strain, which is used in the classical infinitesimal theory of elasticity, is the most common definition applied to materials subjected to very small deformations. However, when deformations are substantial (e.g., elastomers, polymers, and biological tissues), the engineering definition of strain is not applicable and other definitions are often used, such as stretch, logarithmic, or true strain (also called Hencky strain); Green-St. Venant strain; and Almansi-Hamel (Eulerian) strain [15].
It is used as strains in the classical infinitesimal theory of elasticity. The engineering strain is defined as the change in length ΔL per unit of the original length L0 of the material in which the force is being applied. The strain is positive if the material is stretched or negative if compressed. Thus, we have
eE =
∆L . L0
(1)
Stretch ratio The stretch ratio is used in the analysis of materials that show large deformations, such as elastomers. The stretch
Brought to you by | University of Manitoba Authenticated Download Date | 6/9/15 6:08 PM
A. Karimi et al.: Stress-strain definitions for elastic modulus measurement 441
ratio (λ) is the measure of the extensional strain of a differential line element. It is expressed as the ratio of the final length Lf to the initial length L0 of the material line: λ=
Lf
L0
.
(2)
The stretch ratio is related to the engineering strain by eE =
Lf − L0 Lf = − 1 = λ − 1. L0 L0
increments, taking into account the influence of the strain path. Under small strain conditions (typically
Alireza Karimi, Mahdi Navidbakhsh*, Mansour Alizadeh and Reza Razaghi
A comparative study on the elastic modulus of polyvinyl alcohol sponge using different stressstrain definitions Abstract: There have been different stress-strain definitions to measure the elastic modulus of spongy materials, especially polyvinyl alcohol (PVA) sponge. However, there is no agreement as to which stress-strain definition should be implemented. This study was aimed to show how different results are given by the various definitions of stress-strain used, and to recommend a specific definition when testing spongy materials. A fabricated PVA sponge was subjected to a series of tensile tests in order to measure its mechanical properties. Three stress definitions (second Piola-Kichhoff stress, engineering stress, and true stress) and four strain definitions (AlmansiHamel strain, Green-St. Venant strain, engineering strain, and true strain) were used to determine the elastic modulus. The results revealed that the Almansi-Hamel strain definition exhibited the highest non-linear stressstrain relation and, as a result, may overestimate the elastic modulus at different stress definitions (second Piola-Kichhoff stress, engineering stress, and true stress). The Green-St. Venant strain definition failed to address the non-linear stress-strain relation using different definitions of stress and invoked an underestimation of the elastic modulus values. Engineering stress and strain definitions were only valid for small strains and displacements, which make them impractical when analyzing spongy materials. The results showed that the effect of varying the stress definition on the maximum stress measurements was significant but not when calculating the elastic modulus. It is important to consider which stress-strain definition is employed when characterizing *Corresponding author: Mahdi Navidbakhsh, School of Mechanical Engineering, Iran University of Science and Technology, Tehran 16846, Iran, Phone: +00982177209027, Fax: +00982173021585, E-mail: [email protected]; and Tissue Engineering and Biological Systems Research Laboratory, School of Mechanical Engineering, Iran University of Science and Technology, Tehran 16846, Iran Alireza Karimi and Reza Razaghi: Tissue Engineering and Biological Systems Research Laboratory, School of Mechanical Engineering, Iran University of Science and Technology, Tehran 16846, Iran; and School of Mechanical Engineering, Iran University of Science and Technology, Tehran 16846, Iran Mansour Alizadeh: School of Mechanical Engineering, Iran University of Science and Technology, Tehran 16846, Iran
the mechanical properties of spongy materials. Although the true stress-true strain definition exhibits a non-linear relation, we favor it in spongy materials mechanics as it gives more accurate measurements of the material’s response using the instantaneous values. Keywords: Almansi-Hamel strain; polyvinyl alcohol sponge; second Piola-Kirchhoff stress; stress-strain definition; tensile testing. DOI 10.1515/bmt-2013-0110 Received October 16, 2013; accepted March 7, 2014; online first March 29, 2014
Introduction Polyvinyl alcohol (PVA) sponges are currently in widespread use for the removal and management of diffuse fluids/blood at the surgical site [18]. They are also contemplated as the most attractive biomedical polymers owing to a combination of qualities, such as biocompatibility [2, 20, 24, 27], high hydrophilicity [19, 23, 25], excellent mechanical strength and flexibility [17, 19, 20, 25–27], thermal stability and absence of toxicity [21], availability, and relative cheapness [22]. However, the application of this versatile material has been limited to ophthalmic, plastic, and hand surgeries as a biocompatible biodegradable material. Knowledge of the mechanical properties of the PVA sponge may pave the way to find a suitable application in tissue engineering as scaffolding material. Several studies have been conducted to measure the mechanical properties of PVA sponge using biaxial puncture and uniaxial tensile test [12]. They showed beneficial results in terms of stress failure compared with two commercially available sponges. Karimi et al. [11] characterized the mechanical properties of a fabricated PVA sponge (P-sponge) using a uniaxial tensile test for tissue engineering applications. They showed the Young’s modulus and maximum stress of 40 and 9.79 MPa for the PVA sponge, respectively. The Young’s modulus of PVA sponge was measured at different
Brought to you by | University of Manitoba Authenticated Download Date | 6/9/15 6:08 PM
440 A. Karimi et al.: Stress-strain definitions for elastic modulus measurement strain rates to investigate the effects of strain rate on the mechanical properties of spongy materials [6]. The results showed a Young’s modulus of 4.28, 208.33, and 187.51 MPa for the strain rate of 1, 20, and 100 mm/min, respectively. PVA sponges also showed different mechanical properties under longitudinal and circumferential loading directions. The Young’s modulus of the PVA sponge in the longitudinal and circumferential directions was 38.91 and 33.34 MPa, respectively. In addition, the maximum stress in the longitudinal direction was 17.90% greater than that in the circumferential direction [4]. The mechanical behavior of PVA sponge has been shown to be similar to rubber-like materials, such as time-dependent viscoelastic behavior, which can be formulated by a visco-hyper elastic approach under low-strain uniaxial loading [5, 7, 9, 12]. Almost all the measurements on the mechanical properties of spongy materials, especially PVA sponge, have used true stress-true strain or engineering stress-engineering strain definitions. However, there is no agreement about which stress-strain definition should be adopted to determine the mechanical behavior of spongy materials in a tensile testing machine. The stress-strain definition may substantially affect the measured mechanical properties, including elastic modulus, maximum stress, and strain. The main objective of this study is to show the results given by different definitions of stress-strain and to rationalize our recommendation for the use of a specific stress-strain definition to test spongy materials. Three stress definitions (second Piola-Kirchhoff stress, engineering stress, and true stress) and four strain definitions (Almansi-Hamel strain, Green-St. Venant strain, engineering strain, and true strain) are used in this study.
poured into Petri dishes and allowed to stand at room temperature (25–30°C) until cross-linking was completed (48 h).
Materials and methods
Engineering strain
Materials and specimen preparation The preparation of the PVA sponge has been thoroughly described in our previous studies [4, 9]. Briefly, to prepare the PVA aqueous solution, 2 g of PVA (molecular weight = 40,000; Sigma-Aldrich, St. Louis, MO, USA) was dissolved in 100 ml of distilled water at 50°C under stirring at 400 rpm for 6 h. The polymer solution was then cast into cylindrical molds and freeze-dried in order to obtain a PVA spongy matrix. To improve its stability in water, the above sponge was cross-linked by exposure to the vapors of a glutaraldehyde aqueous solution (25%) at 37°C for 24 h. After being rinsed with distilled water, the sponge was freeze-dried again. The final solution was
Axial measurements The initial dimensions of all specimens were measured precisely. The tensile test was performed using a uniaxial tensile test apparatus adapted for testing biological specimens used in our previous studies [8, 10, 14, 15]. All tests were performed at 25°C, and each sample was tested only once. A low strain rate of 5 mm/min, which is typical for surgical procedures and gives more insight into tissue behavior, was employed by the action of an axial servo motor [13]. Adhesive sandpaper of fine grit was placed on the surface of the jaws of the pneumatic grips to prevent slippage. Thereafter, the samples were subjected to a continuous tensile force until failure.
Strain definitions Engineering or conventional strain, which is used in the classical infinitesimal theory of elasticity, is the most common definition applied to materials subjected to very small deformations. However, when deformations are substantial (e.g., elastomers, polymers, and biological tissues), the engineering definition of strain is not applicable and other definitions are often used, such as stretch, logarithmic, or true strain (also called Hencky strain); Green-St. Venant strain; and Almansi-Hamel (Eulerian) strain [15].
It is used as strains in the classical infinitesimal theory of elasticity. The engineering strain is defined as the change in length ΔL per unit of the original length L0 of the material in which the force is being applied. The strain is positive if the material is stretched or negative if compressed. Thus, we have
eE =
∆L . L0
(1)
Stretch ratio The stretch ratio is used in the analysis of materials that show large deformations, such as elastomers. The stretch
Brought to you by | University of Manitoba Authenticated Download Date | 6/9/15 6:08 PM
A. Karimi et al.: Stress-strain definitions for elastic modulus measurement 441
ratio (λ) is the measure of the extensional strain of a differential line element. It is expressed as the ratio of the final length Lf to the initial length L0 of the material line: λ=
Lf
L0
.
(2)
The stretch ratio is related to the engineering strain by eE =
Lf − L0 Lf = − 1 = λ − 1. L0 L0
increments, taking into account the influence of the strain path. Under small strain conditions (typically
