K.Y. Volokh

A model of growth and rupture of abdominal aortic aneurysm

We present here a coupled mathematical model of growth and failure of the abdominal aortic aneurysm (AAA). The failure portion of the model is based on the constitutive theory of softening hyperelasticity where the classical hyperelastic law is enhanced with a new constant indicating the maximum energy that an infinitesimal material volume can accumulate without failure. The new constant controls material failure and it can be interpreted as the average energy of molecular bonds from the microstructural standpoint. The constitutive model is compared to the data from uniaxial tension tests providing an excellent fit to the experiment. The AAA failure model is coupled with a phenomenological theory of soft tissue growth. The unified theory includes both momentum and mass balance laws coupled with the help of the constitutive equations. The microstructural alterations in the production of elastin and remodeling of collagen are reflected in the changing macroscopic parameters characterizing tissue stiffness, strength and density. The coupled theory is used to simulate growth and rupture of an idealized spherical AAA. The results of the simulation showing possible AAA ruptures in growth are reasonable qualitatively while the quantitative calibration of the model will require further clinical observations and in vitro tests. The presented model is the first where growth and rupture are coupled.

Modeling failure of soft anisotropic materials with application to arteries.

The arterial wall is a composite where the preferred orientation of collagen fibers induces anisotropy. Though the hyperelastic theories of fiber-reinforced composites reached a high level of sophistication and showed a reasonable correspondence with the available experimental data they are short of the failure description. Following the tradition of strength of materials the failure criteria are usually separated from stress analysis. In the present work we incorporate a failure description in the hyperelastic models of soft anisotropic materials by introducing energy limiters in the strain energy functions. The limiters provide the saturation value for the strain energy which indicates the maximum energy that can be stored and dissipated by an infinitesimal material volume. By using some popular constitutive models enhanced with the energy limiters we analyze rupture of a sheet of arterial material under the plane stress state varying from the uniaxial to equal biaxial tension. We calculate the local failure criteria including the maximum principal stress, the maximum principal stretch, the von Mises stress, and the strain energy at the moment of the sheet rupture. We find that the local failure criterion in the form of the critical strain energy is the most robust among the considered ones. We also find that the tensile strength-the maximum principal stress-that is usually obtained in uniaxial tension tests might not be appropriate as a failure indicator in the cases of the developed biaxiality of the stress-strain state.

Nonlinear elasticity for modeling fracture of isotropic brittle solids

A softening hyperelastic continuum model is proposed for analysis of brittle fracture. Isotropic material is characterized by two standard parameters-shear and bulk modulus-and an additional parameter of the volumetric separation work. The model can be considered as a volumetric generalization of the concept of the cohesive surface. The meaning of the proposed constitutive equations is clarified by the examples of simple shear and hydrostatic pressure. It is emphasized that the proposed constitutive model includes only smooth functions and the necessary computational techniques are those of nonlinear elasticity.

Comparison of biomechanical failure criteria for abdominal aortic aneurysm

Medical doctors consider a surgery option for the expanding abdominal aortic aneurysm (AAA) when its maximum diameter reaches 5.5cm. This simple geometrical criterion may possibly underestimate the risks of rupture of small aneurysms as well as overestimate the risks of rupture of large aneurysms. Biomechanical criteria of the AAA failure are desired. Various local criteria of the AAA failure are used in the literature though their experimental validation is needed. In the present work, we use the experimentally calibrated AAA model, which includes a failure description, to examine various popular criteria of the local failure. Particularly, we analyze various states of the biaxial tension of the AAA material and evaluate the following criteria of the local failure: (1) the maximum principal stretch; (2) the maximum principal stress; (3) the maximum shear stress; (4) von Mises stress; and (5) the strain energy. The results show that the strain energy is almost constant for the failure states induced by the loads varying from the uniaxial to the equal biaxial tension. The von Mises stress exhibits a wider range of scattering as compared to the strain energy. The maximum stresses and stretches vary significantly with the variation of loads from the uniaxial to the equal biaxial tension.

On the Modified Virtual Internal Bond Method

The virtual internal bond (VIB) method was developed for the numerical simulation of fracture processes. In contrast to the traditional approach of fracture mechanics where stress analysis is separated from a description of the actual process of material failure, the VIB method naturally allows for crack nucleation, branching, kinking, and arrest. The idea of the method is to use atomic-like bond potentials in combination with the Cauchy-Born rule for establishing continuum constitutive equations which allow for the material separation-strain localization. While the conventional VIB formulation stimulated successful computational studies with applications to structural and biological materials, it suffers from the following theoretical inconsistency. When the constitutive relations of the VIB model are linearized for an isotropic homogeneous material, the Poisson ratio is found equal to 1/4 so that there is only one independent elastic constant-Youngs modulus. Such restriction is not suitable for many materials. In this paper, we propose a modified VIB (MVIB) formulation, which allows for two independent linear elastic constants. It is also argued that the discrepancy of the conventional formulation is a result of using only two-body interaction potentials in the microstructural setting of the VIB method. When many-body interactions in bond bending are accounted for, as in the MVIB approach, the resulting formulation becomes consistent with the classical theory of isotropic linear elasticity.

Prediction of Femoral Head Collapse in Osteonecrosis

The femoral head deteriorates in osteonecrosis. As a consequence of that, the cortical shell of the femoral head can buckle into the cancellous bone supporting it. In order to examine the buckling scenario we performed numerical analysis of a realistic femoral head model. The analysis included a solution of the hip contact problem, which provided the contact pressure distribution, and subsequent buckling simulation based on the given contact pressure. The contact problem was solved iteratively by approximating the cartilage by a discrete set of unilateral linear springs. The buckling calculations were based on a finite element mesh with brick elements for the cancellous bone and shell elements for the cortical shell. Results of 144 simulations for a variety of geometrical, material, and loading parameters strengthen the buckling scenario. They, particularly, show that the normal cancellous bone serves as a strong supporting foundation for the cortical shell and prevents it from buckling. However, under the development of osteonecrosis the deteriorating cancellous bone is unable to prevent the cortical shell from buckling and the critical pressure decreases with the decreasing Young modulus of the cancellous bone. The local buckling of the cortical shell seems to be the driving force of the progressive fracturing of the femoral head leading to its entire collapse. The buckling analysis provides an additional criterion of the femoral head collapse, the critical contact pressure. The buckling scenario also suggests a new argument in speculating on the femoral head reinforcement. If the entire collapse of the femoral head starts with the buckling of the cortical shell then it is reasonable to place the reinforcement as close to the cortical shell as possible.

Cavitation instability as a trigger of aneurysm rupture

Aneurysm formation and growth is accompanied by microstructural alterations in the arterial wall. Particularly, the loss of elastin may lead to tissue disintegration and appearance of voids or cavities at the micron scale. Unstable growth and coalescence of voids may be a predecessor and trigger for the onset of macroscopic cracks. In the present work, we analyze the instability of membrane (2D) and bulk (3D) voids under hydrostatic tension by using two experimentally calibrated constitutive models of abdominal aortic aneurysm enhanced with energy limiters. The limiters provide the saturation value for the strain energy, which indicates the maximum energy that can be stored and dissipated by an infinitesimal material volume. We find that the unstable growth of voids can start when the critical stress is considerably less than the aneurysm strength. Moreover, this critical stress may even approach the arterial wall stress in the physiological range. This finding suggests that cavitation instability can be a rational indicator of the aneurysm rupture.

On Irreversibility and Dissipation in Hyperelasticity With Softening

Bulk and interface material failures are often modeled via hyperelastic stored energy functions incorporating softening behavior. The softening is reversible due to the hyperelastic nature of the constitutive law and material can “heal” under unloading. To prevent this healing, special numerical procedures (like finite element deletion) are usually used in computer simulations. In the present work, we suggest an alternative: very simple analytical formulation, which makes failure irreversible when a critical stored energy is reached. This new notion is directly incorporated into the constitutive equations, consequently, relieving the need for preliminary discretization of the boundary-value problem. [DOI: 10.1115/1.4026853]

An approach to elastoplasticity at large deformations

While elastoplasticity theories at small deformations are well established for various materials, elastoplasticity theories at large deformations are still a subject of controversy and lively discussions. Among the approaches to finite elastoplasticity two became especially popular. The first, implemented in the commercial finite element codes, is based on the introduction of a hypoelastic constitutive law and the additive elastic–plastic decomposition of the deformation rate tensor. Unfortunately, the use of hypoelasticity may lead to a nonphysical creation or dissipation of energy in a closed deformation cycle. In order to replace hypoelasticity with hyperelasticity the second popular approach based on the multiplicative elastic–plastic decomposition of the deformation gradient tensor was developed. Unluckily, the latter theory is not perfect as well because it introduces intermediate plastic configurations, which are geometrically incompatible, non-unique, and, consequently, fictitious physically. In the present work, an attempt is made to combine strengths of the described approaches avoiding their drawbacks. Particularly, a tensor of the plastic deformation rate is introduced in the additive elastic–plastic decomposition of the velocity gradient. This tensor is used in the flow rule defined by the generalized isotropic Reiner-Rivlin fluid. The tensor of the plastic deformation rate is also used in an evolution equation that allows calculating an elastic strain tensor which, in its turn, is used in the hyperelastic constitutive law. Thus, the present approach employs hyperelasticity and the additive decomposition of the velocity gradient avoiding nonphysical hypoelasticity and the multiplicative decomposition of the deformation gradient associated with incompatible plastic configurations. The developed finite elastoplasticity framework for isotropic materials is specified to extend the classical J2-theory of metal plasticity to large deformations and the simple shear deformation is analyzed.

Modeling rupture of growing aneurysms

Growth and rupture of aneurysms are driven by micro-structural alterations of the arterial wall yet precise mechanisms underlying the process remain to be uncovered. In the present work we examine a scenario when the aneurysm evolution is dominated by turnover of collagen fibers. In the latter case it is natural to hypothesize that rupture of individual fibers (or their bonds) causes the overall aneurysm rupture. We examine this hypothesis in computer simulations of growing aneurysms in which constitutive equations describe both collagen evolution and failure. Failure is enforced in constitutive equations by limiting strain energy that can be accumulated in a fiber. Within the proposed theoretical framework we find a range of parameters that lead to the aneurysm rupture. We conclude in a qualitative agreement with clinical observations that some aneurysms will rupture while others will not.