Hypoelastic soft tissues. Part I: Theory

Refinements are made to an existing hypoelastic theory developed by Freed [18, 19] for the purpose of modeling the passive response of soft, fibrous, biological tissues. Oldroyd’s [27] operators for convected differentiation and integration, which he derived from the tensor transformation law, are re-derived here from an integral equation defined in the polar configuration. Fields that obey these convected polar operators are said to be viable tensor fields, from which a new definition for strain and its rate are obtained and applied to a hypoelastic theory for tissue. Anisotropy is addressed through a material tensor, from which viable tensor fields describing fiber strain and strain rate are constructed. Material anisotropy and material constitution are handled separately for maximum flexibility. Isochoric hypoelastic models for isotropic, anisotropic, and fiber/matrix composite materials are derived. A material function is introduced to address special attributes that biological fibers impart on tissue behavior, four of which are proposed that represent various ways through which the fiber constituents might be described. Application to in-plane biaxial deformation is the focus of part II of this paper [23].     

Analysis of the mechanical behavior of woven fibrous material using virtual tests at the unit cell level

The determination of the mechanical properties of fabrics in biaxial tension and in-plane shearing is made from 3D finite element analyses of the unit woven cell. Compared to experimental tests these virtual tests have several advantages. They can easily be carried out for sets of varied parameters, they provide local information inside the woven material and above all they can be performed on woven materials that have not yet been manufactured. The 3D computations are not classical analyses because the yarns are made up of several thousands of fibres and their mechanical behaviour is very special. Several specific aspects of the analysis are detailed, especially the use of a hypoelastic law based on an objective derivative using the rotation of the fibre which allows a strict evolution of the directions of orthotropy according to the fibre direction. Examples of analyses are presented in biaxial tension and in-plane shear for woven reinforcements and in the case of the biaxial tension of a knitted fabric. The results obtained are in good agreement with experimental results.     

Post-buckling analysis of skew plates subjected to combined in-plane loadings

Post-buckling behavior of laminated composite, sandwich and functionally graded skew plates is analyzed in the present work. The problem formulation is based on higher-order shear deformation theory and von Kármán’s nonlinear kinematics. Linear mapping is used to transform the physical domain into the computational domain. Chebyshev polynomials are used for spatial discretization of governing differential equations and boundary conditions. The nonlinear terms are linearized using quadratic extrapolation technique. The effect of the skew angle on the buckling and post-buckling response of the composite, sandwich and FGM-clamped skew plates is investigated for different combinations of in-plane compressive loadings.     

An inverse method for material parameters determination of titanium samples under tensile loading

Different approaches used for the simulation of woven reinforcement forming are investigated. Especially several methods based on finite element approximation are presented. Some are based on continuous modelling, while others, called discrete or mesoscopic approaches, model the components of the fabric. A semi discrete finite element made of woven unit cells under biaxial tension and in-plane shear is detailed. In continuous approaches, the difficulty lies in the necessity to take the strong specificity of the fibrous material into account. The yarn directions must be strictly followed during the large strains of the fabric. This is the main goal of the non-orthogonal model and of the hypoelastic constitutive model based on the yarn rotation presented in this paper. In the case of discrete and semi-discrete approaches the directions of the yarns are “naturally” followed because the yarns are modeled. Explicitly, however, modeling each component at the mesoscopic scale can lead to high numerical cost.     

Fracture behaviour by two cracks around an elliptic rigid inclusion

An analysis is presented for the in-plane biaxial loading of an elastic system consisting of a partially bonded rigid elliptic inclusion embedded in an infinite matrix where the bond imperfections are two symmetric cracks. The boundary value problem is formulated through the complex variable technique in the case of linearly elastic matrix under general biaxial loading at infinity. The elastic solution is obtained for a simplified model by assuming incompressibility and plane strain conditions for the matrix. In particular, stress and displacement fields are represented and discussed.

Moreover, a stress criterion, which permits to take into account, either the crack extension at the interface or its deviation into the matrix, is considered to study the fracture response of the elastic system.     

Characterising the shear–tension coupling and wrinkling behaviour of woven engineering fabrics

Modelling the forming process for engineering fabrics and textile composites using a mechanical approach, such as the finite element method, requires characterisation of the material’s behaviour under large shear deformation. For woven engineering fabrics, a coupling between in-plane tension and both shear compliance and the onset of wrinkling is to be expected. This paper focuses on a novel testing technique, the biaxial bias extension test, as a means to investigate this shear–tension coupling and fabric wrinkling. Novel methods of determining the wrinkling behaviour are demonstrated. The main difficulty with the technique lies in extracting the material contribution to the recorded signal. To do this, an experimental method is proposed and demonstrated using a plain weave glass fabric. Biaxial bias extension test results are compared against picture frame and uniaxial bias extension results.     

CALCULATION OF HYPERELASTIC RESPONSE OF FINITELY DEFORMED ELASTIC-VISCOPLASTIC MATERIALS

The objective of this paper is to present a numerical algorithm for calculating hyperelastic constitutive equations characterizing the thermomechanical response of elastically isotropic elastic-viscoplastic materials. The algorithm is developed within the context of an alternative formulation of plasticity in which elastic distortional deformation is determined directly by integrating an evolution equation which includes the current velocity gradient and quantities that depend only on the present state of the material. Consequently, the formulation is independent of the particular choice of a measure of plastic deformation, the reference configuration, and the total deformation gradient from the reference configuration. These features allow the constitutive equations to be easily implemented into computer codes which currently use a hypoelastic formulation for calculating plasticity.     

The plastic spin concept and a simple illustration of its role in finite plastic transformations

The basic kinematics and the rate constitutive equations are briefly presented within the framework of a macroscopic formulation of finite plastic transformations for structured media, employing the concept of tensorial structure variables. The key feature is the necessity to provide constitutive relations for the plastic spin according to the proposition of Mandel and Kratochvil. After obtaining these relations for a transversely isotropic medium employing the representation theorems for isotropic second order antisymmetric tensor-valued functions, the role of the plastic spin is illustrated by the analysis of large simple shear for such a medium.     

Topology optimization of finite strain viscoplastic systems under transient loads

A transient finite strain viscoplastic model is implemented in a gradient‐based topology optimization framework to design impact mitigating structures. The model's kinematics relies on the multiplicative split of the deformation gradient, and the constitutive response is based on isotropic hardening viscoplasticity. To solve the mechanical balance laws, the implicit Newmark‐beta method is used together with a total Lagrangian finite element formulation. The optimization problem is regularized using a partial differential equation filter and solved using the method of moving asymptotes. Sensitivities required to solve the optimization problem are derived using the adjoint method. To demonstrate the capability of the algorithm, several protective systems are designed, in which the absorbed viscoplastic energy is maximized. The numerical examples demonstrate that transient finite strain viscoplastic effects can successfully be combined with topology optimization.     

An elastic-viscoplastic model exhibiting continuity of solid and fluid states

In this paper specific unified constitutive equations for an elastic-viscoplastic material are developed which in the limit of infinite resistance to plastic flow become those of a general hyperelastic isotropic material; and in the limit of zero resistance to plastic flow become those of a general Reiner-Rivlin fluid. The constitutive equations satisfy the restrictions imposed by continuum thermodynamics and are hyperelastic in the sense that the symmetric Piola-Kirchhoff stress is related to a derivative of the Helmholtz free energy. In this formulation the Helmholtz free energy depends on five scalar invariants: two new scalars which are pure measures of “elastic” distortional deformation, a measure of total dilatation, a measure of plastic dilatation, and temperature. Specifically, the stress is not characterized by a hypoelastic equation so no special rates of stress need be considered. Both isotropic hardening and a measure of directional hardening which models the Bauschinger effect are included. Finally, examples of simple shear are considered to examine the solid-like and fluid-like response to large deformations including cycles of loading, unloading and reloading.     

Stability analysis of FGM sandwich plates by using variable-kinematics Ritz models

Abstract

A linearized buckling analysis of functionally graded material (FGM) isotropic and sandwich plates is carried out by virtue of the Hierarchical Trigonometric Ritz Formulation (HTRF). Quasi-3D Ritz models based on equivalent single layer (ESL) and zig zag (ZZ) plate theories are developed within the framework of the Carrera Unified Formulation (CUF). Several in-plane loading conditions accounting for axial, biaxial, and shear loadings are taken into account. Parametric studies are carried out in order to evaluate the effects of significant parameters, such as volume fraction index, length-to-thickness ratio, sandwich plate type, and loading type, on the critical buckling loads.     

Closed-form integration of singular terms for constant,linear and quadratic boundary elements. Part 2. SV-P wave propagation

Part I of these two related papers considered the analytical evaluation of singular integrals for anti-plane boundary elements, the results of which were then applied to the evaluation of scattering problems involving SH waves. This second part provides an extension of these results to the more complicated case of in-plane boundary elements, and presents their application to scattering problems involving SV-P waves. First, the singular integrals for constant, linear and quadratic boundary elements are evaluated a in closed form. Thereafter, the formulation is used to model cylindrical inclusions in a two-dimensional elastic medium illuminated by dynamic in-plane (plane-strain) line sources. The boundary element method (BEM) results are then compared with the known analytical solutions for these problems.     

Continuously deforming finite elements

A general class of time-dependent co-ordinate transformations is introduced in a variational formulation for evolution problems. The variational problem is posed with respect to both solution and transformation field variables. An approximate analysis using finite elements is developed from the continuous variational form. Modified forms of the variational functional are considered to ensure the deforming mesh is not top irregular. ODE system integrators are utilized to integrate the resulting semidiscrete systems. In Part I we consider the formulation for problems in one spatial dimension and time, including, in particular, convection-dominated flows described by the convection-diffusion, Burgers' and Buckley–Leverett equations. In Part II the extension of the method to two dimensions and supporting numerical experiments are presented.     

On the non-singular traction-BIE in elasticity

The work reported herein develops a generalized traction-BIE formulation which involves only weakly singular integrals (in the three-dimensional problem) or totally regular integrals (in the two-dimensional problem). The first step deals with the terms in the Somigliana displacement identity, and then the derivatives of these terms. The only conditions required for the existence of the traction-BIE and the related Somigliana stress identity are weak continuity of the in-plane derivatives of the surface displacements and of the surface tractions. It is shown that the Cauchy Principal Value (CPV) interpretations so commonly used in BIE developments are unnecessary. The formulation is established not only at a smooth boundary point, but also at a corner point. The extension of the non-singular formulation to discontinuous boundary tractions and tangential derivatives of the boundary displacements applicable to a generalized problem statement as well as the usual BEM implementations is also shown. In the demonstrated formulation, the source points are located directly at the boundary nodes and non-conformal elements are not needed.     

Non-symmetric extension of a crack

Summary The dynamic in-plane problem of the non-symmetric extension of a crack in an infinite, isotropic elastic medium under normal stress is analyzed. Following Cherepanov [8], Cherepanov and Afanas'ev [9] the general solution of the problem is derived in terms of an analytic function of complex variable. The results include the expressions for the stress intensity factors at the crack tips and the rate of energy flux into the cxtending crack edges. For a particular case, numerical calculations for the stress intensity factor and the energy flux rate are carried out.     

Non-singular term effect on the fracture quantities of a crack in a piezoelectric medium

The static problem of a crack in a piezoelectric plate subjected to biaxial loading at infinity is analyzed. The aim of this paper is to estimate the influence of non-singular terms originated by the load biaxiality on the stress fields and on the elastic and electric displacements in the vicinity of the crack tip. An analytical method for seeking the electro-elastic solution is proposed. The novel procedure involves a transformation of similarity induced by the fundamental matrix that enables to express the equations governing the problem in terms of complex potentials. The application of the boundary conditions leads then to the formulation of Hilbert problems whose solutions allow to obtain the generalized stress and displacement components. Numerical results and graphs are presented and discussed for various loading conditions. The non-singular solution is compared to the asymptotic one, generally considered in the literature when analyzing fracture problems. In particular, it is shown that the direction of incipient crack extension, sought through the maximum circumferential stress criterion, can be seen to deviate from the crack axis as the collinear load increases, although geometry and applied load are symmetric.     

Biaxial Yield for Nonlinearly Viscoelastic Materials with a Strain Clock

A constitutive equation with a dilatation dependent reduced time is used to model the mechanical response of solid amorphous polymers such as polycarbonate. Such constitutive equations have the property that stress relaxation occurs faster with increasing dilatation. In previous work, it has been shown that this constitutive equation can account for yield in materials undergoing uniaxial strain or stress control histories. In the present work, yield is discussed when materials described by this constitutive equation undergo homogeneous biaxial and triaxial strain histories. Four sets of conditions are considered: in-plane biaxial constant strain rate histories and in-plane biaxial constant stress rate histories, for both plane stress and plane strain states. Yield is defined in a manner analogous to that in the corresponding strain and stress control conditions in the uniaxial case.     

Interface flaw extension under in-plane loadings

The wave propagation problem associated with symmetric extension of a stress-free crack at a constant rate along the interface of perfectly bonded elastic half-spaces under uniform in-plane loadings is considered. Because no characteristic length appears in the problem formulation, homogeneous function techniques are applied and approximate solutions are obtained as single integrals of complex functions. As in the corresponding static problem, the values of all the field variables undergo rapid oscillations near the crack edges. This behavior can be interpreted as interpenetration of the half-space materials in a small region near the crack edges.     

Stress around square and rectangular cutouts in symmetric laminates

The solution presented in this paper is useful for finding the stress distribution around holes in symmetric laminates as well as in isotropic plates and also to determine the failure strength of the laminate on first ply failure basis by Tsai-Hill, Hashin-Rotem and Tsai-Wu criteria. This is a one stop solution for all kinds of in-plane loading on symmetric laminates as well as isotropic plates with any shape of cutout. Using Savin’s basic solution for anisotropic plates, the stress functions are derived for generalized mapping function for the hole and arbitrarily oriented in-plane loading. Square and rectangular holes in symmetric laminates of Graphite/epoxy and Glass/epoxy are studied. It is noted that the maximum stress and its location is mainly influenced by the type of loading. Larger stresses are noted for shear loading. The stress results are also obtained by ANSYS for comparison.