A mixed Von Mises distribution for modeling soft biological tissues with two distributed fiber properties

Constitutive modeling of biological tissues plays an important role in the understanding of tissue behavior and the development of synthetic materials for medical and bio-inspired applications. A structural continuum model that incorporates principal structural features of the tissue can potentially provide the link between microstructure and the macroscopic mechanical response of biological tissues. For most soft biological tissues, including arterial walls and skin tissue, the main load-carrying constituent is presumed to be the distributed collagen fibers embedded in a base matrix. It is believed that the organization of the collagen fibers gives rise to the anisotropy of the material. In this paper, a semi-structural constitutive model is proposed to account for planar fiber distributions with more than one distributed planar fiber property. Motivated by histology information of the wing membrane of the bat, a statistical treatment is formulated in this paper to capture the overall effect of the distribution of fiber cross-sectional area and the distribution of the number of fibers. This formulation is suitable for general cases when more than one fiber property varies spatially. Furthermore, this model is a two-dimensional specialization within the framework of a three-dimensional theory, which is different the formulation based on a fundamentally two-dimensional theory.     

A Micromechanics Finite-Strain Constitutive Model of Fibrous Tissue

Biological tissues have unique mechanical properties due to the wavy fibrous collagen and elastin microstructure. In inflation, a vessel easily distends under low pressure but becomes stiffer when the fibers are straightened to take up the load. The current microstructural models of blood vessels assume affine deformation, i.e., the deformation of each fiber is assumed to be identical to the macroscopic deformation of the tissue. This uniform-field (UF) assumption leads to the macroscopic (or effective) strain energy of the tissue that is the volumetric sum of the contributions of the tissue components. Here, a micromechanics-based constitutive model of fibrous tissue is developed to remove the affine assumption and to take into consideration the heterogeneous interactions between the fibers and the ground substance. The development is based on the framework of a recently developed second-order homogenization theory, and takes into account the waviness, orientations and spatial distribution of the fibers, as well as the material nonlinearity at finite-strain deformation. In an illustrative simulation, the predictions of the macroscopic stress-strain relation and the statistical deformation of the fibers are compared to the UF model, as well as finite-element (FE) simulation. Our predictions agree well with the FE results, while the UF predictions significantly overestimate. The effects of fiber distribution and waviness on the macroscopic stress-strain relation are also investigated. The present mathematical model may serves as a foundation for native as well as for engineered tissues and biomaterials.     

A structural constitutive model for smooth muscle contraction in biological tissues

A structural constitutive model that characterizes the active and passive responses of biological tissues with smooth muscle cells (SMCs) is proposed. The model is formulated under the assumption that the contractile units in SMCs and the connected collagen fibers are the active tissue component, while the collagen fibers not connected to the SMCs are the passive tissue component. An evolution law describing the deformation of the active tissue component over time is developed based on the sliding filament theory. In this evolution law the contraction force is the sum of a motor force that initiates contraction, a viscous force that describes the actin–myosin filament sliding, and an elastic force that accounts for the deformation of the cross-bridges. The mechanical response of the collagen fibers is governed by the fiber recruitment process: collagen fibers support load and behave as a linear elastic material only after becoming taut. The proposed structural constitutive model is tested with published active and passive, uniaxial and biaxial experimental data on pig arteries.     

A structural multi-mechanism constitutive equation for cerebral arterial tissue

A structural multi-mechanism constitutive equation is developed to describe the nonlinear, anisotropic, inelastic mechanical behavior of cerebral arterial tissue. Elastin and collagen fibers are treated as separate components (mechanisms) of the artery. Elastin is responsible for load bearing at low strain levels while the collagen mechanism is recruited for load bearing at higher strain levels. This work builds on an earlier model in which both the elastin and collagen mechanisms are represented by isotropic response functions [Wulandana, R., Robertson, A.M., 2005. An inelastic multi-mechanism constitutive equation for cerebral arterial tissue. Biomech. Model. Mechan. 4 (4), 235–248]. Here, the anisotropic material response of the wall is introduced through the collagen mechanism which is composed of helically distributed families of fibers. The orientation of these families is described using either a finite number of fiber orientations or a fiber distribution function. The fiber orientation or dispersion function can be prescribed directly from arterial histology data, or, taking a phenomenological approach, based on data fitting from bi-axial measurements. The activation of the collagen mechanism is specified using a new fiber strain based activation criterion. The multi-mechanism constitutive equation is applied to the simple case of cylindrical inflation and material constants are determined based on available inelastic experimental data for cerebral arteries. While the proposed model captures all features of this inelastic data, there is a pressing need for further experiments to refine the model.     

A stochastic-structurally based three dimensional finite-strain damage model for fibrous soft tissue

A fully three dimensional finite-strain damage model for fibrous soft tissue is developed. The model assumes uncoupled contributions for the matrix and collagen fibers, and uncoupled bulk and deviatoric response over any range of deformations. A simple isotropic damage mechanism within the framework of continuum damage mechanics has been used to describe the softening behavior under deformation for the matrix. On the other hand, statistical aspects related to the length distribution of the reinforcing fibers lead to a damage model for the reinforcing material. As a result, a general theoretical framework for constitutive modeling of biological soft tissue with continuum damage is obtained. A theoretical example consisting of a biaxial test of a soft tissue reinforced with two families of collagen fibers has been considered to demonstrate the capabilities of the proposed model and to study the sensitivity to changes in the statistical parameters associated with the reinforcing material. Also, a preliminary numerical example is included to demonstrate the model on a inhomogeneous boundary value problem. Results show that the model is able to capture the typical stress-strain behavior observed in fibrous soft tissue and seems to confirm the soundness of the proposed formulation.     

The effect of fiber recruitment on the swelling of a pressurized anisotropic non-linearly elastic tube

We study the effect of fiber recruitment on the mechanical response of a fiber reinforced non-linearly elastic tube that is both swollen and pressurized. Attention is restricted to cylindrically symmetric tube deformation. The constitutive model permits fibers to support tension, but not compression. While many combinations of pressure and swelling cause all of the fibers to be recruited for load support, both large swelling and large deswelling can give rise to fiber derecruitment at certain locations in the tube. This leads to less channel opening than would be the case if the fibers provided support while contracted. The transition between mechanically active and mechanically inactive fibers can be described in terms of the quasi-static motion of a fiber recruiting interface.     

Continuous versus discrete (invariant) representations of fibrous structure for modeling non-linear anisotropic soft tissue behavior

The non-linear anisotropic mechanical response of soft tissue is largely dependent on the structure of the underlying collagen network. Collagen structure has been successfully quantified for various tissue types in terms of a locally defined fiber orientation distribution function. The continuous distribution function derived from structural data can be directly incorporated into an integral representation of the strain energy function for modeling tissue behavior. Alternatively, non-integral (often invariant-based) strain energy functions have been developed in which the collagen network structure is approximated using a discrete set of fiber classes. The advantage of such an approach is increased computational efficiency since the values of the strain energy and its derivatives (e.g. stress) can be evaluated without numerical integration. However, because of the structural simplifications such models are presumably unable to predict mechanical data as accurately as the models which incorporate a continuous orientation distribution function. In this work the ability of discrete versus continuous fiber models to capture the non-linear anisotropic response of soft tissue is critically analyzed. Both unimodal and bimodal fiber distributions are considered. A general formulation has been developed in terms of an arbitrary fiber strain energy function, such that the analysis can be performed for any suitable fiber material model. For tissue structures in which a discrete representation is suitable, techniques are presented for establishing the range of loading conditions in which model accuracy is not significantly compromised, thus justifying the use of an invariant-based modeling approach.     

Elastic–plastic behavior of non-woven fibrous mats

Electrospinning is a novel method for creating non-woven polymer mats that have high surface area and high porosity. These attributes make them ideal candidates for multifunctional composites. Understanding the mechanical properties as a function of fiber properties and mat microstructure can aid in designing these composites. Further, a constitutive model which captures the membrane stress–strain behavior as a function of fiber properties and the geometry of the fibrous network would be a powerful design tool. Here, mats electrospun from amorphous polyamide are used as a model system. The elastic–plastic behavior of single fibers are obtained in tensile tests. Uniaxial monotonic and cyclic tensile tests are conducted on non-woven mats. The mat exhibits elastic–plastic stress–strain behavior. The transverse strain behavior provides important complementary data, showing a negligible initial Poisson's ratio followed by a transverse:axial strain ratio greater than ?1:1 after an axial strain of 0.02. A triangulated framework has been developed to emulate the fibrous network structure of the mat. The micromechanically based model incorporates the elastic–plastic behavior of single fibers into a macroscopic membrane model of the mat. This representative volume element based model is shown to capture the uniaxial elastic–plastic response of the mat under monotonic and cyclic loading. The initial modulus and yield stress of the mat are governed by the fiber properties, the network geometry, and the network density. The transverse strain behavior is linked to discrete deformation mechanisms of the fibrous mat structure including fiber alignment, fiber bending, and network consolidation. The model is further validated in comparison to experiments under different constrained axial loading conditions and found to capture the constraint effect on stiffness, yield, post-yield hardening, and post-yield transverse strain behavior. Due to the direct connection between microstructure and macroscopic behavior, this model should be extendable to other electrospun systems and other two dimensional random fibrous networks.     

An elastic–plastic crack bridging model for brittle-matrix fibrous composite beams under cyclic loading

A fibrous composite beam with an edge crack is submitted to a cyclic bending moment and the crack bridging actions due to the fibers. Assuming a general elastic-linearly hardening crack bridging model for the fibers and a linear-elastic law for the matrix, the statically indeterminate bridging actions are obtained from compatibility conditions. The elastic and plastic shake-down phenomena are examined in terms of generalised cross-sectional quantities and, by employing a fatigue crack growth law, the mechanical behaviour up to failure is captured. Within the framework of the proposed fracture mechanics-based model, the cyclic crack bridging due to debonding at fiber–matrix interface of short fibers is analysed in depth. By means of some simplifying assumptions, such a phenomenon can be described by a linear isotropic tensile softening/compressive hardening law. Finally, numerical examples are presented for fibrous composite beams with randomly distributed short fibers.     

On some connections between equivalent single material and mixture theory models for fiber reinforced hyperelastic materials

There are two approaches that can be used to model the large strain mechanical response of material systems in which elastic fibers are embedded in an elastic matrix. In the first approach, a fiber reinforced material undergoing large deformation is homogenized in the sense that it is assumed to act as an equivalent single material that is transversely isotropic and hyperelastic. Both constituents then share a common reference configuration, which is typically assumed to be a natural or stress-free configuration for the equivalent single material. The stress depends on a single deformation gradient defined with respect to the natural configuration.In the second approach, the fiber/matrix system is treated as a mixture, with the matrix and the fibrous constituents having their own reference configurations and material symmetries. The total stress depends on the deformation gradients and material symmetries for both constituents, defined with respect to their reference configurations.Under appropriate assumptions, the constitutive theory developed using mixture theory can coincide with the constitutive theory assuming an equivalent single material that is transversely isotropic and hyperelastic. This paper explores the connection between the two approaches by considering the various reference configurations and material symmetries.     

Homogenization of long fiber reinforced composites including fiber bending effects

This paper presents a homogenization method, which accounts for intrinsic size effects related to the fiber diameter in long fiber reinforced composite materials with two independent constitutive models for the matrix and fiber materials. A new choice of internal kinematic variables allows to maintain the kinematics of the two material phases independent from the assumed constitutive models, so that stress–deformation relationships, can be expressed in the framework of hyper-elasticity and hyper-elastoplasticity for the fiber and the matrix materials respectively. The bending stiffness of the reinforcing fibers is captured by higher order strain terms, resulting in an accurate representation of the micro-mechanical behavior of the composite. Numerical examples show that the accuracy of the proposed model is very close to a non-homogenized finite-element model with an explicit discretization of the matrix and the fibers.     

Mechanics Applied to Skeletal Ontogeny and Phylogeny

The musculo-skeletal system serves the mechanical function of creating motion and transmitting loads. It is made up mainly of four components: bone, cartilage, muscle and fibrous connective tissue. These have evolved over millions of years into the complex and diverse shapes of the animal skeleton. The skeleton, however, is not built to a static plan: it can adapt to mechanical forces during growth, it can remodel if the forces change, and it can regenerate if it is damaged. In this paper, the regulation of skeletal construction by mechanical forces is analyzed from both ontogenetic and phylogenetic standpoints. In the first part, models of biomechanical processes that act during skeletal ontogenesis – tissue differentiation and bone remodeling – are presented and, in the second, the evolution of the middle ear is used as an example of biomechanical change in skeletal phylogenesis. Because the constitutive laws for skeletal tissues are relatively well understood, and because the skeleton is preserved in the fossil record, application of mechanics to skeletal evolution seems to present a good opportunity to explore the relationships governing ontogenetic adaptations and phylogenetic change.     

On non-physical response in models for fiber-reinforced hyperelastic materials

Soft biological tissues are sometimes composed of thin and stiff collagen fibers in a soft matrix leading to a strong anisotropy. Commonly, constitutive models for quasi-incompressible materials, as for soft biological tissues, make use of an additive split of the Helmholz free-energy into a volumetric and a deviatoric part that is applied to the matrix and fiber contribution. This split offers conceptual and numerical advantages. The purpose of this paper is to investigate a non-physical effect that arises thereof. In fact, simulations involving uniaxial stress configurations reveal volume growth at rather small stretches. Numerical methods such as the Augmented Lagrangian method might be used to suppress this behavior. An alternative approach, proposed here, solves this problem on the constitutive level.     

A constitutive model for the in-plane mechanical behavior of nonwoven fabrics

A constitutive model is presented for the in-plane mechanical behavior of nonwoven fabrics. The model is developed within the context of the finite element method and provides the constitutive response for a mesodomain of the fabric corresponding to the area associated to a finite element. The model is built upon the ensemble of three blocks, namely fabric, fibers and damage. The continuum tensorial formulation of the fabric response rigorously takes into account the effect of fiber rotation for large strains and includes the nonlinear fiber behavior. In addition, the various damage mechanisms experimentally observed (bond and fiber fracture, interfiber friction and fiber pull-out) are included in a phenomenological way and the random nature of these materials is also taken into account by means of a Monte Carlo lottery to determine the damage thresholds. The model results are validated with recent experimental results on the tensile response of smooth and notched specimens of a polypropylene nonwoven fabric.     

Predictive nonlinear constitutive relations in polymers through loss history

A model is presented that calculates the highly nonlinear mechanical properties of polymers as a function of temperature, strain and strain rate from their molecular structure. The model is based upon the premise that mechanical properties are a direct consequence of energy stored and energy dissipated during deformation of a material. This premise is transformed into a consistent set of structure–property relations for the equation of state and the engineering constitutive relations in a polymer by quantifying energy storage and loss at the molecular level of interactions between characteristic groups of atoms in a polymer. The constitutive relations are formulated as a set of analytical equations that predict properties directly in terms of a small set of structural parameters that can be calculated directly and independently from the chemical composition and morphology of a polymer.     

A composites-based hyperelastic constitutive model for soft tissue with application to the human annulus fibrosus

This paper presents a composites-based hyperelastic constitutive model for soft tissue. Well organized soft tissue is treated as a composite in which the matrix material is embedded with a single family of aligned fibers. The fiber is modeled as a generalized neo-Hookean material in which the stiffness depends on fiber stretch. The deformation gradient is decomposed multiplicatively into two parts: a uniaxial deformation along the fiber direction and a subsequent shear deformation. This permits the fiber-matrix interaction caused by inhomogeneous deformation to be estimated by using effective properties from conventional composites theory based on small strain linear elasticity and suitably generalized to the present large deformation case. A transversely isotropic hyperelastic model is proposed to describe the mechanical behavior of fiber-reinforced soft tissue. This model is then applied to the human annulus fibrosus. Because of the layered anatomical structure of the annulus fibrosus, an orthotropic hyperelastic model of the annulus fibrosus is developed. Simulations show that the model reproduces the stress-strain response of the human annulus fibrosus accurately. We also show that the expression for the fiber-matrix shear interaction energy used in a previous phenomenological model is compatible with that derived in the present paper.     

Polarized Image Correlation for Large Deformation Fiber Kinematics

The underlying fiber architecture of soft tissues, like bat wing skin, plays an important role in the material’s overall mechanical behavior. The mesoscopic birefringent fiber architecture of the bat wing skin can be visualized directly under polarized light. This inherent property is the key to the new experimental technique developed in this work: polarized image correlation (PIC). PIC is a technique for determining full field material strains and fiber kinematics of mesoscopically resolved fibrous tissues during biaxial mechanical testing. Not only is the material birefringence used to determine fiber kinematics under finite deformations, but it is also used for image correlation and strain field computation. Pure translation tests performed with PIC verify the accuracy of the technique. A segmental image processing method was developed to solve an experimental issue of changing birefringent properties to construct accurate continuous deformation profiles. By integrating PIC with traditional digital image correlation, both surface and subsurface data give additional insight into through thickness tissue behavior. The PIC technique is applicable to semi-transparent tissues with birefringent mesosopic structures; incorporation of microscopy would resolve smaller fiber structures. Future work includes extending the techniques to three dimensions to analyze curved surfaces.     

In Situ Mechanical Testing of Hydrated Biological Nanofibers Using a Nanoindenter Transducer

Nanoscale biological fibers, such as collagen, keratin or elastin, serve as building blocks for a wide variety of biological tissues (for example, bone, skin and hair). As such, the elasticity, strength and damage tolerance of these fibers largely control the mechanical performance of tissues at the macroscale. While there is a large body of experimental data for tests on whole biological tissues at the macroscale, mechanical tests on individual biological fibers are scarcer because of their small size (400?nm diameter or less). Isolating, imaging, handling and testing these fibers in hydrated conditions are significant challenges. The AFM-based and MEMS-based techniques developed in the past to test such fibers offer high displacement and load resolution, but they lack the stroke and force capability required to fracture strong and highly extensible fibers such as collagen fibrils. In this work, a microscale mechanical testing platform capable of measuring the tensile stress?Cstrain response of individual type I collagen fibers and fibrils was developed and validated. The platform is composed of a capacitance-based, nanoindenter transducer, an optical microscope to monitor the deformation of the sample in situ and a set of micromanipulators to isolate and handle individual fibers and fibrils. Our preliminary results on type I collagen demonstrate the feasibility of monotonic and cyclic tensile tests under the optical microscope and in hydrated conditions. The setup can be used to study the elasticity, strength and damage tolerance of type I collagen fibers and fibrils (using cyclic tests), and our preliminary data are consistent with existing experiments and predictions from numerical models. This setup offers the advantage of being composed from relatively standard components (optical microscope, nanoindenter) which means that it can be easily duplicated in laboratories that already possess these instruments. This technique can be used to assess the effect of environment, genetic diseases or therapeutic drugs on biological tissues at a fundamental level.