Compressive Mechanical Properties of Porcine Brain: Experimentation and Modeling of the Tissue Hydration Effects
Abstract
:1. Introduction
2. Materials and Methods
2.1. Sample Preparation
2.2. Testing Apparatuses
2.2.1. Mach-1™ for Quasi-static Testing of the Wet Brain
2.2.2. Split-Hopkinson Pressure Bar (SHPB) for High Strain Rate Testing of Wet and Dry Brain Specimens
2.2.3. Instron™5568 for Quasi-Static Testing of the Dry Brain
2.3. Stress–Strain Experimental Data
2.4. Statistical Analysis of the Experimental Data
2.5. Finite Element Simulation-Based Micromechanics of the Dry Brain and Water
3. Results and Discussion
3.1. Experiment Response
3.2. Simulation Response
4. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Appendix A
Appendix B
Appendix C
Symbol | Description |
---|---|
Wave velocity | |
Particle velocity | |
, s | Wave velocity and particle velocity linear relationship constants |
Free energy | |
Rate of change of the strain energy density function | |
Elastic part or the right Cauchy–Green tensor | |
,, | Internal strain fields (internal state variables) |
Evolving strain threshold | |
, | Saturation value of , |
Cauchy stress | |
F | Deformation gradient |
, | Elastic part of F, transpose of the elastic part of F |
, | Plastic part of F, transpose of the plastic part of F |
Determinant of F, determinant of , Determinant of | |
Elastic rate of deformation, inelastic rate of deformation | |
Second Piola–Kirchoff stress | |
Plastic rotational deformation | |
Elastic portion of the rotation tensor, transpose of the elastic portion of the rotation tensor | |
Right stretch tensor | |
Mandel stress | |
Kirchoff stress (elasto-viscoplastic part) | |
Elastic shear moduli | |
Identity matrix | |
Elastic part of the Green–Lagrange strain | |
Bulk moduli | |
, | Stress-like thermodynamic conjugates to , |
Stress-like thermodynamic conjugate to | |
Viscous shear strain rate | |
Reference viscous shear strain rate | |
Effective pressure | |
Equivalent shear stress | |
αp | Stress-like thermodynamic conjugate of |
Y | Yield criterion |
Stretch tensor | |
Rate of stretch tensor | |
, , | Hardening moduli |
Intermediate plastic configuration | |
σt | Elastic-viscoelastic transition stress |
σp | Elastic-plastic transition stress |
Direction of viscous flow | |
Hoppy bar velocities | |
t | Time |
Strain | |
, | Hoppy bar sample strain rate, hoppy bar sample strain |
Sample instantaneous length | |
Hoppy bar areas | |
Hoppy bar forces | |
Hoppy bar elastic moduli | |
Strain in the incident and transmitted hoppy bars | |
Hoppy bar sample stress | |
Hoppy bar sample area | |
Engineering stress, engineering strain | |
Initial sample area | |
P | Force applied on the material |
Current and reference lengths | |
True stress, true strain |
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Strain Rate (s−1) | Wet/Dry | Number of Samples | Number of Animals/Porcine Brains | Temperature (°C) | Pressure (MPa) |
---|---|---|---|---|---|
0.00625 | Wet | 6 | 3 | 20.85 | 0.1 |
0.025 | Wet | 7 | 4 | 20.85 | 0.1 |
0.1 | Wet | 16 | 7 | 20.85 | 0.1 |
50 | Wet | 6 | 3 | 20.85 | 0.1 |
250 | Wet | 4 | 2 | 20.85 | 0.1 |
450 | Wet | 5 | 2 | 20.85 | 0.1 |
550 | Wet | 7 | 3 | 20.85 | 0.1 |
750 | Wet | 4 | 2 | 20.85 | 0.1 |
0.00625 | Dry | 4 | 2 | 20.85 | 0.1 |
0.1 | Dry | 5 | 3 | 20.85 | 0.1 |
250 | Dry | 4 | 2 | 20.85 | 0.1 |
Total | 68 | 33 |
FE Simulation Case | Dry Brain % (m/m) | Water % (m/m) |
---|---|---|
1 | 80 | 20 |
2 | 60 | 40 |
3 | 40 | 60 |
4 | 20 | 80 |
Term/Function | Description |
---|---|
, where is the elastic part of the right Cauchy-Green tensor, and , and are internal strain fields (internal state variables). | Free energy, |
, where Je-1 is the inverse of the determinant of . and are the elastic part of F and the transpose of elastic part of F. | Cauchy Stress, Second Piola-Kirchhoff Stress, |
, where and are the elastic part of the rotation tensor (R) and the transpose of the elastic part of R. , where μ and K are the elastic shear and bulk moduli modeling the elastic behavior respectively. is the elastic part of the Green-Lagrange strain tensor, and is the identity matrix. , where is the plastic part of the deformation tensor, and is the right stretch tensor. | Kirchhoff Stress (elasto-viscoplastic part, ) Elastic Law (Mandel Stress, ) Deformation Gradient |
, , where and are stress-like thermodynamic conjugates of the and respectively. , where is a stress-like thermodynamic conjugate of . | Stress-like internal state variables Stress-like internal state variable |
, where is the inelastic rate of deformation. with , where is the viscous shear strain rate given by the following equation: with and , where is an equivalent shear stress term and is the effective pressure term, is a reference strain rate, m is a strain rate sensitivity parameter, and αp is a pressure sensitivity parameter and Y is the yield criterion. , , where represents an evolving strain threshold or criterion that the macromolecular chains must overcome to slip. h0 and g0 are hardening moduli, and is the saturation value of . with and where is the hardening modulus, and is the saturation value of ξ2. and | Flow rule Equivalent plastic shear strain-rate Polymer chain resistance to plastic flow Polymer chain crystallization at large strain Evolution equation of stretch-like tensor |
Material constants |
Model Constants | Constant Definition | Values |
---|---|---|
μ (MPa) | Shear Modulus | 0.80 |
K (MPa) | Bulk Modulus | 399.73 |
(s−1) | Reference Strain Rate | 120,000 |
m | Strain Rate Sensitivity Parameter | 0.90 |
Yo (MPa) | Material Yield Parameter | 9.00 |
αp | Sensitivity Parameter | 0 |
λL | Network Locking Stretch | 2.00 |
μR | Rubbery Modulus | 0.07 |
Rs1 | Material Hardening Parameter | 1.4 |
ho | Hardening Modulus | 0.41 |
ξo1 | Internal Strain-Like Parameter Initial Value | 0.0045 |
ξ*sat | Internal Strain-Like Parameter Saturation Value | 0.001 |
ξ*o | Energetic Strain Barrier | 1.2 |
go | Hardening Modulus | 0.3 |
Cκ1 (MPa) | Internal Stress-Like Parameter | 0.41 |
h1 | Hardening Modulus | 0 |
eos2 | Internal Strain-Like Parameter Initial Value | 0 |
esats2 | Internal Strain-Like Parameter Saturation Value | 0.4 |
Cκ2 (MPa) | Internal Stress-Like Parameter | 0 |
Strain Rate (s−1) | 0.00625 s−1 n = 6 | 0.0250 s−1 n = 7 | 0.100 s−1 n = 16 | p-Value | |
---|---|---|---|---|---|
Variable | |||||
Tangent Modulus (kPa) | 1.6822 ± 0.0047 | 1.7497 ± 0.0021 | 3.2378 ± 0.0371 | <0.05 | |
Transition Stress (kPa) | 0.1046 ± 0.0046 | 0.1069 ± 0.0020 | 0.1400 ± 0.0377 | >0.05 | |
Strain at Peak Stress | 0.075 ± 0.0040 | 0.075 ± 0.0060 | 0.072 ± 0.0050 | >0.05 |
Strain Rate (s−1) | 0.00625 s−1 n = 4 | 0.100 s−1 n = 5 | 250 s−1 n = 4 | p-Value | |
---|---|---|---|---|---|
Variable | |||||
Tangent Modulus (kPa) | 2591.451 ± 424.4080 | 2282.160 ± 922.3810 | 2143.683 ± 620.0000 | >0.05 | |
Transition Stress (kPa) | 108.602 ± 17.7860 | 138.313 ± 55.9020 | 166.389 ± 48.1420 | >0.05 | |
Strain at Transition Stress | 0.0419 ± 0.0086 | 0.0606 ± 0.0100 | 0.0776 ± 0.0106 | >0.05 |
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Prabhu, R.K.; Begonia, M.T.; Whittington, W.R.; Murphy, M.A.; Mao, Y.; Liao, J.; Williams, L.N.; Horstemeyer, M.F.; Sheng, J. Compressive Mechanical Properties of Porcine Brain: Experimentation and Modeling of the Tissue Hydration Effects. Bioengineering 2019, 6, 40. https://doi.org/10.3390/bioengineering6020040
Prabhu RK, Begonia MT, Whittington WR, Murphy MA, Mao Y, Liao J, Williams LN, Horstemeyer MF, Sheng J. Compressive Mechanical Properties of Porcine Brain: Experimentation and Modeling of the Tissue Hydration Effects. Bioengineering. 2019; 6(2):40. https://doi.org/10.3390/bioengineering6020040
Chicago/Turabian StylePrabhu, Raj K., Mark T. Begonia, Wilburn R. Whittington, Michael A. Murphy, Yuxiong Mao, Jun Liao, Lakiesha N. Williams, Mark F. Horstemeyer, and Jianping Sheng. 2019. "Compressive Mechanical Properties of Porcine Brain: Experimentation and Modeling of the Tissue Hydration Effects" Bioengineering 6, no. 2: 40. https://doi.org/10.3390/bioengineering6020040
APA StylePrabhu, R. K., Begonia, M. T., Whittington, W. R., Murphy, M. A., Mao, Y., Liao, J., Williams, L. N., Horstemeyer, M. F., & Sheng, J. (2019). Compressive Mechanical Properties of Porcine Brain: Experimentation and Modeling of the Tissue Hydration Effects. Bioengineering, 6(2), 40. https://doi.org/10.3390/bioengineering6020040