J. Sebastian Giudice, Lee F. Gabler, Matthew B....
1
J. Sebastian Giudice, [email protected] 4040 Lewis and Clark Drive Charlottesville, VA 22911 University of Virginia - Center for Applied Biomechanics Model for Human Heel Pad Under High Rate Axial Loading A Multi-Stage Inverse Finite Element Method for Determining the Constitutive J. Sebastian Giudice, Lee F. Gabler, Matthew B. Panzer Introduction Previous analysis using a Finite Element (FE) model of the leg has demonstrated that the nonlinear and visco- elastic properties of the human heel pad significantly influence the force transmitted to the tibia in high-rate axial compression [1]. Suitable closed-form analytical solutions do not exist for determining the constitutive relationship of the materials tested, due to complex boundary conditions, sample dimensions, and inertial effects. Inverse Finite Element analysis (iFE) can be used to determine constitutive models from mechanical testing with no analytical solutions, but is limited by the large computational costs associated with optimization. Methods Results Discussion Objectives Constitutive Model Conclusions The multi-stage QLV method provided a fit to the experimental data for heel pad comparable to those previously generated using an analytical method [1] When the analytically-derived QLV constitutive model was applied to the FE model, the force response was over-predicted. The multi-staged approach reduced the computational cost and the SSE by factors of 2.3 and 4.8, respectively when compared to the single-stage optimization * Varied significantly with accuracy of initial guesses. Best case reported. The multi-stage iFE method was sensitive to initial guesses for the model parameters 1. Develop a multi-stage iFE method where nonlinear and viscoelastic material parameters are optimized for individual tests over their effective time domains 2. Apply this method to a FE representation of the heel pad to determine its Quasi-Linear Viscoelastic (QLV) constitutive model 3. Investigate differences in the constitutive model for iFE and previously determined analytical methods [1]. Multi-Stage iFE Stage 1 : The IER parameters (μ 0 , γ) are optimized over a short duration simulation. Assumed no relaxation and all G i = 0. t = 10 ms. Stage 2 : The first RRF parameter (G 1 ) is included in the optimization and all other G i = 0. t = 15 ms. Stage 3 : The second RRF parameter (G 2 ) is included in the optimization and all other G i = 0. t = 30 ms. Stage 4 : The third RRF parameter (G 3 ) is included in the optimization and all other G i = 0. t = 100 ms. Stage 5 : The final RRF parameter (G 4 ) is included in the optimization. t = 1000 ms. 1. A multi-stage inverse Finite Element method was developed and applied to a heel pad FE model to determine its QLV constitutive model. 2. An analytical method provided a similar fit to the IFE method, however, performed poorly when implemented into the FE heel pad model 3. The multi-stage iFE method provided a better fit to the experimental data than the traditional iFE method, with less computational cost 4. The novel multi-stage iFE method can be applied to any viscoelastic material that exhibits a strain-dependent initial elastic response and time-dependent stress relaxation Future Work Perform iFE analysis on remaining cases to establish a characteristic model and statistical significance. Implement the iFE heel pad characteristic model into the GHBMC leg and assess its performance in high-rate axial compression of the tibia [1,2]. Validate the heel pad characteristic model in other loading conditions (i.e. cyclic) to further assess the robustness of the multi-stage iFE method. [1] Gabler, Lee F., Panzer, Matthew B., Salzar, Robert S. High-Rate Mechanical Properties of Human Heel Pad for Simulation of a Blast Loading Condition. Proc. Int. Res. Counc. Biomech. Impact IRCOBI 796–80 (2014). [2] Henderson K, Bailey A, Christopher J, Brozoski F, Salzar R. Biomechanical response of the lower leg under high rate loading. Proceedings of IRCOBI Conference, 2013, Gothenburg, Sweden. [3] Fung YC, Biomechanics: Mechanical Properties of Living Tissues, 277-280, Springer, New York, NY, 1993. 0 0.2 0.4 0.6 0.8 1 1 10 100 1000 10000 Time (ms) Unscaled RRF Terms τ = 1 τ = 10 τ = 100 τ = 1000 − Instantaneous Elastic Response (IER) Reduced Relaxation Function (RRF) , = − ′ ( ′ ⅆ ′ = ∞ + = ∙ − =1ms = s = s =1000ms Strain Energy Density (SED) Quasilinear Viscoelasticity Framework [1] = ∞ + = ≤ ≤ Constraints Ramp and Hold Tests • 17 Cubic samples • = (10-45)% Compression • Avg./Peak = 25s -1 /60s -1 FE Model • Heel pad quarter-model • 1024 hexahedral elements • Incompressible material • Platen • Single rigid hexahedral element Uniaxial Compression • Experimental displacement- time history prescribed to platen 0.0 0.1 0.2 0.3 1 10 100 1000 10000 Eng. Strain Time (ms) Input Strain-Time History ε = [ − − ] = + − ( − The stress response , to finite strain ( was modeled using a QLV framework [1,3]. =− iFE Method SSE Calculation Time (hrs) Traditional 3.1* 17.8* Multi-Stage 0.6 7.57 Fit and CPU Time for iFE Methods References Heel Pad Platen Model accuracy assessed using the sum of squared error (SSE). = = − 0 1 2 3 0 5 10 Force (N) Time (ms) Stage 1 Optimization Results Experimental Stage 1 0 1 2 0 5 10 15 Force (N) Time (ms) Stage 2 Optimization Results Experimental Stage 2 0 1 2 0 10 20 30 Force (N) Time (ms) Stage 3 Optimization Results Experimental Stage 3 0 1 2 0 50 100 Force (N) Time (ms) Stage 4 Optimization Results Experimental Stage 4 Stage G 1 G 2 G 3 G 4 G inf 1 31.7 3 0 0 0 0 1 2 34.1 4.8 0.8 0 0 0 0.2 3 34.9 5 0.8041 0.1036 0 0 0.0923 4 34.9 5 0.8041 0.0995 0.0492 0 0.0473 5 34.9 5 0.8041 0.0995 0.0437 0.018 0.0347 iFe Material Model Convergence Analytical and iFE Material Models QLV Model Coefficients Analytical Values [1] iFE Values (kPa) 31.43 35.00 10.86 5.000 (τ₁ = 1ms) 0.7246 0.8041 (τ₂ = 10ms) 0.1633 0.0995 (τ₃ = 100ms) 0.0454 0.0437 (τ₄ = 1000ms) 0.0179 0.0180 ∞ (τ→∞) 0.0355 0.0347 SSE 0.7378 0.6428 t = 0 ms t = 100 ms t = 10 ms t = 1000 ms = 0 = 14 = 3 = 8 The analytically determined constitutive model was applied to the FE heel pad model under identical boundary conditions 0 0.5 1 1.5 2 1 10 100 1000 Force (N) Time (ms) FE Force Response with iFE and Analytically Derived Material Models Experimental iFE Analytical Analytical (FE)
Transcript of J. Sebastian Giudice, Lee F. Gabler, Matthew B....
J. Sebastian Giudice, [email protected] Lewis and Clark Drive
Charlottesville, VA 22911University of Virginia - Center for Applied Biomechanics
Model for Human Heel Pad Under High Rate Axial LoadingA Multi-Stage Inverse Finite Element Method for Determining the Constitutive
J. Sebastian Giudice, Lee F. Gabler, Matthew B. Panzer
Introduction Previous analysis using a Finite Element (FE) model of
the leg has demonstrated that the nonlinear and visco-elastic properties of the human heel pad significantly influence the force transmitted to the tibia in high-rate axial compression [1].
Suitable closed-form analytical solutions do not exist for determining the constitutive relationship of the materials tested, due to complex boundary conditions, sample dimensions, and inertial effects.
Inverse Finite Element analysis (iFE) can be used to determine constitutive models from mechanical testing with no analytical solutions, but is limited by the large computational costs associated with optimization.
Methods
Results Discussion
Objectives
Constitutive Model
Conclusions
The multi-stage QLV method provided a fit to the experimental data for heel pad comparable to those previously generated using an analytical method [1]
When the analytically-derived QLV constitutive model was applied to the FE model, the force response was over-predicted.
The multi-staged approach reduced the computational cost and the SSE by factors of 2.3 and 4.8, respectively when compared to the single-stage optimization
* Varied significantly with accuracy of initial guesses. Best case reported.
The multi-stage iFE method was sensitive to initial guesses for the model parameters1. Develop a multi-stage iFE method where nonlinear and
viscoelastic material parameters are optimized for individual tests over their effective time domains
2. Apply this method to a FE representation of the heel pad to determine its Quasi-Linear Viscoelastic (QLV) constitutive model
3. Investigate differences in the constitutive model for iFEand previously determined analytical methods [1].
Multi-Stage iFEStage 1: The IER parameters (μ0, γ) are optimized over a short duration
simulation. Assumed no relaxation and all Gi = 0. t = 10 ms.Stage 2: The first RRF parameter (G1) is included in the optimization and
all other Gi = 0. t = 15 ms.Stage 3: The second RRF parameter (G2) is included in the optimization
and all other Gi = 0. t = 30 ms.Stage 4: The third RRF parameter (G3) is included in the optimization and
all other Gi = 0. t = 100 ms.Stage 5: The final RRF parameter (G4) is included in the optimization.
t = 1000 ms.
1. A multi-stage inverse Finite Element method was developed and applied to a heel pad FE model to determine its QLV constitutive model.
2. An analytical method provided a similar fit to the IFE method, however, performed poorly when implemented into the FE heel pad model
3. The multi-stage iFE method provided a better fit to the experimental data than the traditional iFE method, with less computational cost
4. The novel multi-stage iFE method can be applied to any viscoelastic material that exhibits a strain-dependent initial elastic response and time-dependent stress relaxation
Future Work
Perform iFE analysis on remaining cases to establish a characteristic model and statistical significance.
Implement the iFE heel pad characteristic model into the GHBMC leg and assess its performance in high-rate axial compression of the tibia [1,2].
Validate the heel pad characteristic model in other loading conditions (i.e. cyclic) to further assess the robustness of the multi-stage iFE method.
[1] Gabler, Lee F., Panzer, Matthew B., Salzar, Robert S. High-Rate Mechanical Properties of Human Heel Pad for Simulation of a Blast Loading Condition. Proc. Int. Res. Counc. Biomech. Impact IRCOBI 796–80 (2014).[2] Henderson K, Bailey A, Christopher J, Brozoski F, Salzar R. Biomechanical response of the lower leg under high rate loading. Proceedings of IRCOBI Conference, 2013, Gothenburg, Sweden.[3] Fung YC, Biomechanics: Mechanical Properties of Living Tissues, 277-280, Springer, New York, NY, 1993.
0
0.2
0.4
0.6
0.8
1
1 10 100 1000 10000
Time (ms)
Unscaled RRF Terms
τ = 1
τ = 10
τ = 100
τ = 1000
𝑒−𝑡 𝜏
Instantaneous Elastic Response (IER)
Reduced Relaxation Function (RRF)
𝝈 𝜺, 𝒕 =
𝟎
𝒕
𝑮 𝒕 − 𝒕′ 𝛛𝝈𝒆(𝜺
𝛛𝜺
𝛛𝜺
𝛛𝒕′ⅆ𝒕′
𝑮 𝒕 = 𝑮∞ +
𝒊=𝟏
𝟒
𝑮𝒊 ∙ 𝒆−
𝒕𝝉𝒊
𝝉𝟏 =1ms𝝉𝟐 = 𝟏𝟎𝐦s
𝝉𝟑 = 𝟏𝟎𝟎𝐦s𝝉𝟒 =1000ms
Strain Energy Density (SED)
Quasilinear Viscoelasticity Framework [1]
𝟏 = 𝑮∞ +
𝒊=𝟏
𝟒
𝑮𝒊 𝟎 ≤ 𝑮𝒊 ≤ 𝟏
Constraints
Ramp and Hold Tests• 17 Cubic samples • 𝜀 = (10-45)% Compression• Avg./Peak 𝜀 = 25s-1/60s-1
FE Model• Heel pad quarter-model
• 1024 hexahedral elements
• Incompressible material• Platen
• Single rigid hexahedral element
Uniaxial Compression• Experimental displacement-
time history prescribed to platen
0.0
0.1
0.2
0.3
1 10 100 1000 10000
En
g. S
trai
n
Time (ms)
Input Strain-Time History
ε
𝑾 =𝝁
𝟐𝜷[𝒆𝜷 𝑰𝟏−𝟑 − 𝟏]
𝝈𝒆 𝝀 =𝝁𝒐𝒆
𝜸 𝝀𝟐+𝟐𝝀−𝟑
(𝝀𝟑 − 𝟏
𝝀𝟑
The stress response 𝜎 𝜀, 𝑡 to finite strain 𝜀(𝑡 was modeled using a QLV framework [1,3].
𝜺 = 𝝀 − 𝟏
iFE Method SSE Calculation Time (hrs)
Traditional 3.1* 17.8*
Multi-Stage 0.6 7.57
Fit and CPU Time for iFE Methods
References
Heel Pad
Platen
Model accuracy assessed using the sum of squared error (SSE).
𝑺𝑺𝑬 =
𝒊=𝟏
𝑵
𝒚𝒊 − 𝒇 𝒙𝒊𝟐
0
1
2
3
0 5 10
Fo
rce
(N)
Time (ms)
Stage 1 Optimization Results
Experimental Stage 1
0
1
2
0 5 10 15
Fo
rce
(N)
Time (ms)
Stage 2 Optimization Results
Experimental Stage 2
0
1
2
0 10 20 30
Fo
rce
(N)
Time (ms)
Stage 3 Optimization Results
Experimental Stage 3
0
1
2
0 50 100
Fo
rce
(N)
Time (ms)
Stage 4 Optimization Results
Experimental Stage 4
Stage 𝝁𝟎 𝜸 G1 G2 G3 G4 Ginf
1 31.7 3 0 0 0 0 12 34.1 4.8 0.8 0 0 0 0.23 34.9 5 0.8041 0.1036 0 0 0.09234 34.9 5 0.8041 0.0995 0.0492 0 0.04735 34.9 5 0.8041 0.0995 0.0437 0.018 0.0347
iFe Material Model Convergence
Analytical and iFE Material Models
QLV Model Coefficients
Analytical Values [1]
iFE Values
𝝁𝟎 (kPa) 31.43 35.00
𝜸 10.86 5.000
𝑮𝟏(τ₁ = 1ms) 0.7246 0.8041
𝑮𝟐(τ₂ = 10ms) 0.1633 0.0995
𝑮𝟑(τ₃ = 100ms) 0.0454 0.0437
𝑮𝟒 (τ₄ = 1000ms) 0.0179 0.0180
𝑮∞(τ→∞) 0.0355 0.0347
SSE 0.7378 0.6428t = 0 ms t = 100 mst = 10 ms t = 1000 ms
𝜎 = 0 𝑘𝑃𝑎𝜎 = 14 𝑘𝑃𝑎 𝜎 = 3 𝑘𝑃𝑎𝜎 = 8 𝑘𝑃𝑎
The analytically determined constitutive model was applied to the FE heel pad model under identical boundary conditions
0
0.5
1
1.5
2
1 10 100 1000
Forc
e (N
)
Time (ms)
FE Force Response with iFE and Analytically Derived Material Models
Experimental iFE Analytical Analytical (FE)
