Constitutive models for polymers and soft biological tissues

I would like to thank the Office of Naval Research (ONR) for their invaluable financial support of this work. I would like to thank Tomonora Honda, Rick Burns, and Vaughan Thomas for the great times we had together at Caltech.

Introduction

Several sets of experimental data are compared to model predictions, demonstrating the ability of the model to reproduce the observed behavior of polymers and soft biological tissues. The order of presentation of topics in the thesis is as follows: Chapter 2 presents the formulation of the constitutive model of polymers and soft biological tissues.

Model formulation

Free energy

Ogden-type hyperelasticity

Cv is the specific heat per unit mass at constant volume and T0 is the reference temperature. The internal variables Fp,Z andFvi,Zvi are closely related through appropriate differential equations or flow rules that will be introduced later. the logarithmic elastic strain measurements.

Deviatoric and volumetric plasticity

• Deviatoric plasticity
• Volumetric plasticity

The initial void volume fraction of the body in the undeformed configuration is given by. Neglecting the elastic volume change of the voids, the plastic volumetric deformation can be expressed as a function of the void radius.

Thermodynamic forces

Evolution laws—Rate effects

Microinertia

Variational formulation of the rate problem

The functionalΦ[ ˙ϕ,Z˙p,Mp,Np,Z˙vi,Mvi,j] does not depend on spatial derivatives of its fields, therefore the minimization (2.59) can be obtained locally as. The Euler-Lagrange equations of the power functionalΦ with respect to Z˙p and Z˙vi are the kinetic relations (2.48), and its Euler-Lagrange equations with respect to Mp, Np and Mvi,j lead to the optimal directions of plastic and viscous flow, as indicated in subsection 2.8.

Incremental constitutive updates

The symmetry of the consistent tangent is a direct consequence of the potential structure of the incremental problem.

Predictor-corrector implementation

The symmetry of the consistent tangent is a direct consequence of the potential structure of the increasing problem. where, j are also the eigenvectors of! e, prei, k+1, and follow from co-linearity betweenMpand! E, preK+1 (cf. [94]), proper optimization with respect to viscous flow directions, j (cf. [28]), and assumption of zero-incremental plastic and viscous deformations in the predictive phase. The optimization is not related to Mp,Npyet, after some algebraic manipulations. 2.77) definesMp implicitly, which can be expressed as.

Stress update

Validation

• Parameter identification
• High strain rate compression tests on polyurea
• Tension tests on polyurea
• Tensile tests on high-density polyethylene
• Monotonic and cyclic uniaxial tests on brain tissue

If assigned, numerical simulations of the experiments can be used to obtain a set of predictions. The worst individuals are removed to maintain a constant population size. Each individual has a probability of being selected equal to the individual's ability divided by the sum of the abilities of all individuals in the population.

It is worth noting that the selection strength for a steady GA is twice that of a generational GA, where for every m members of the population there are only 2 mselections. 3.7) One set of parameters was obtained for all 15 experiments, proving the outstanding efficiency of the GAs approach. This shows the ability of the model to distinguish between elastic and plastic volumetric deformation (strain rate: !˙= 0.001s−1).

A first set of parameter estimates was obtained through a viscoelastic fitting model (ve) ignoring the plastic part of the equilibrium network. The positive {µ, α} pairs in tension (Ne = 2), negative pairs in compression (Ne = 3) and significantly higher shear moduliµ0,µ∞in compression confirm the well-known notion in the literature about the heterogeneous tensile-compressive character of brain tissue (cf.

Figure 3.1: Alireza et al. experimental confined compression curves [5] vs. GAs fits. Fitting curves correspond to one set of material parameters (cf

Application to ballistic and blast impact on composite plates and shells

Introduction

Also, the ballistic properties of flax, hemp and jute fabric reinforced polypropylene composites processed by hot compression molding were investigated by Wambua et al. The ballistic effect of the composites was investigated by examining the ballistic limit of hybrid composite-steel systems fabricated by bonding thin mild steel plates to the front and back sides of the natural fiber composites [92]. It was found that the ballistic properties of the hemp composites increased significantly when a mild steel plate was used as a cladding and backing [92].

The ballistic properties of Kevlar 29/Polivnyl Butyral and polyethylene fiber composites used in the light armor design were analyzed experimentally and numerically by Colakoglu et al. In the following sections, the ballistic impact of a high-velocity projectile on a polyurea retrofitted DH36 steel plate is investigated . The formulation of the contact potential used to model the impact forces is summarized in section 4.2.2.

Ballistic impact on composite plates

• Localization elements
• Modeling contact forces
• Composite plate shot experimental setup
• Validation

The kinematics of the strain localization elements are identical to the kinematics of cohesive elements proposed by Ortiz and Pandolfi [73] for the simulation of fracture. The potential energy of the body has contributions from the tension energy from the bulk. The existence of voids in polyurea was observed in the microstructure of the spall region (Fig. 4.5), which required the activation of volumetric plasticity in the model.

Contact was also used to capture the effect of the reflected wave in the kite as it passes through the interface and reaches the original wave in the target plate, resulting in spalling, as shown in figure. The simulated normal velocity of the free target surface is found to be in good agreement with the experimental results (see Figure 4.7). The model outlined in this work was used for polyurea where the activation of thermal softening allowed the formation of shear bands (thermal properties obtained from [45] and Primeaux Associates LLC are shown in table 4.4.) The parameters obtained for polyurea in section 3.3 and Tab.

Figure 4.2: Naval Surface Warfare Center (Dahlgren Division) Research Gas Gun Facility (Courtesy of Bill Mock et al

Blast impact on composite shells

• Subdivision thin-shell elements
• Shell fracture and fragmentation
• DH36 steel/Polyurea composite hull
• Aluminum/PVC foam H100/Aluminum composite hull
• PVC foam - Divinycell H100

Two simulations were performed; the first with a composite hull consisting of 0.0048mm DH36 steel and 0.01056mm polyurea coating on the opposite side of. Another composite hull configuration involves the use of Divinycell H100, a PVC foam with a density of 100 Kg/m3, in the central part of the hull of the thickness laminates (see Fig. 4.20). All the values ​​characterizing the response of the Divinycell H100 foam used in the following analysis are reported in Tab, 4.6, where σc is the compressive stress at which the slope of the stress-strain curve first changes.

The ultimate compressive strength, −50 MPa, is one of the lowest available values ​​for the compressive strength of solid PVC [2]. In the presented calculations, the first stress peak is used as the compressive collapse of the foam [59]. The loading conditions impose bending on the structure, so the stress/compression material properties of the PVC foam were appropriately assigned to the thickness integration points.

Figure 4.14: One cohesive edge and the two adjacent subdivision shell elements with their one-neighborhoods

Application to brain trauma

Introduction

The development of coup and/or contra coup lesions also depends on which part of the skull is affected. TBI is also produced by rotational movements of the cranial parenchyma (angular acceleration injuries) and flexion-extension of the craniospinal junction (Adams and Graham [3]). DAI also results in different and widespread areas of the brain no longer being able to function or intercommunicate.

Gliar cellular responses cause axonal retraction balls and/or microglial scarring or fiber tract degeneration. This chapter covers the biomechanical modeling of brain tissue response to advancing impact waves, and the computational simulation of traumatic brain injury. Potential applications of the current research to relevant medical and engineering problems are explored in the concluding chapter.

Figure 5.1: Coup-contrecoup injury (adapted from Kleiven [46])

Finite element model of the human head

In contrast to other elastic and viscoelastic approaches available in the literature, the present model is able to reproduce permanent brain tissue damage, in the form of plastic sliding between brain layers and irreversible growth of voids or bubbles in the material, which simulate the effects. of DAI and cavitation injury. The model also includes time-dependent viscous deformations and large perturbations of the material from thermodynamic equilibrium, via an exact finite viscoelasticity theory with validation against available experimental results on brain tissue samples in Chapter 3. The ability of the present theory to reproduce real tissue damage mechanisms is illustrated , and predictions of intracranial pressure, shear stress, cavitation, and shear injury dynamics are presented.

The entire model consists of 39047 tetrahedral composite elements [89] and is characterized by a level of detail comparable to that of the Wayne State Brain Injury Model (Zhou et al. [102]). Viscoelastic material properties commonly used in the literature for head injury simulations (cf. Horgan and Gilchrist [42]) were appropriately adapted to the current model. The volumetric viscositiesηvoli were set to zero, assuming purely elastic volumetric behavior in the viscoelastic networks.

Figure 5.2: Mid-sagittal and mid-coronal sections of the adopted head finite element model:

Impact simulations

• Frontal impact
• Oblique impact

In line with the parietal lobe, a pressure profile similar in shape to that of the frontal region, but with reduced amplitude, was observed (Figure 5.5). Contour curves of intracranial pressure on the mid-sagittal section of the head model are shown in the figure. The contour curves of the viscous shear strain !v in the first viscoelastic mechanism are shown in fig.

It is noticeable that τ profiles follow the contours of the velocity gradient associated with the rotational motion of the head (Fig. 5.11). The spikes initially appear below the cortical surface and then evolve towards the core areas of the brain (cf. [101]). Therefore, the occurrence of DAI can be predicted evolving from the periphery to the core of the brain.

Figure 5.3: Frontal impact injury

Conclusions

• Summary
• Outlook
• Polymeric applications
• Medical applications
• Neuromuscular applications
• A concluding remark

There is a wide range of uses for computational modeling of soft materials such as polymers and biological tissues. In addition, studies have shown the power of computational modeling in predicting polymer properties, particularly at the interface and in solution. Thus, computational modeling of polymers can provide a valuable tool to complement the ongoing research on the interface between phase-separated domains.

In retrospect, the lack of computer modeling status limits the usefulness of surgical simulation. Less than half a century old, continuous computer modeling of soft materials is undoubtedly still in its infancy, and its predicted prospects remain promising for improving the quality of life for mankind. Soft materials modeling plays a key role in developing the necessary mathematical models and analyses.

Bibliography

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A rate-independent elastoplastic constitutive model for biofiber-reinforced composites at finite strains: a continuum basis, algorithmic formulation, and finite element implementation. Mechanics of rate-dependent elastic-plastic deformation of glassy polymers from low to high strain rates. Effect of projectile mass and velocity on brain response to cannon impact.