Cauchy-Green Strain Tensor Research Articles

SU-E-J-161: Biomechanical Framework for Thoracic Tumors Characteristics.

Respiration-induced kinematics of thoracic tumors suggests a simple analogy with elasticity, where a strain is used to characterize the volume of interests. The application of the biomechanical framework allows for the objective determination of tumor characteristics. The deformation of a given tissue element can be determined if its displacement is known. The latter can be obtained from 4DCT scans using image registration of the end of inhalation and exhalation CT volumes. The averaged right Cauchy-Green strain tensor was determined for each of the 15 retrospectively analyzed thoracic GTVs. The departure of the strain tensor from the identity matrix gauges the departure of the medium from rigidity. The averaging was carried out in Log-Euclidean framework. The fractional and geodesic anisotropy factors were determined for the tensor. The amplitude of GTV motion varied from 0.64 to 4.21 with the average of 1.2cm. The GTV size ranged from 5.16 to 149.99cc with the average of 43.19cc. The tumor deformation is inconsiderable and the tensorial anisotropy is small. The Log-Euclidean distance of averaged strain tensors from the identity matrix ranged from 0.06 to 0.31 with the average of 0.19. The Frobenius distance from the identity matrix is similar and ranged from 0.06 to 0.35 with the average of 0.21. Their fractional anisotropy ranged from 0.02 to 0.12 with the average of 0.07. Their geodesic anisotropy ranged from 0.03 to 0.16 with the average of 0.09. These values also indicate insignificant deformation. The biomechanical framework allows for the quantitative description of the tissue or anatomical regions of interest. Such regional characterization of volume of interests can be used in the objective evaluation of changes of the anatomy during the treatment or after the treatment. It might be used for correlation of outcome studies with objective characterization of changes within biomechanical framework. These objective characteristics do not rely on human interpretation. The measured changes might have predictive characteristics for the therapeutic success of the treatment.

Computing Lagrangian coherent structures from their variational theory

Using the recently developed variational theory of hyperbolic Lagrangian coherent structures (LCSs), we introduce a computational approach that renders attracting and repelling LCSs as smooth, parametrized curves in two-dimensional flows. The curves are obtained as trajectories of an autonomous ordinary differential equation for the tensor lines of the Cauchy-Green strain tensor. This approach eliminates false positives and negatives in LCS detection by separating true exponential stretching from shear in a frame-independent fashion. Having an explicitly parametrized form for hyperbolic LCSs also allows for their further in-depth analysis and accurate advection as material lines. We illustrate these results on a kinematic model flow and on a direct numerical simulation of two-dimensional turbulence.

A note on the Mooney–Rivlin material model

In finite elasticity, the Mooney–Rivlin material model for the Cauchy stress tensor T in terms of the left Cauchy–Green strain tensor B is given by $$T = -pI + s_1 B + s_2 B^{-1},$$ where p is the pressure and s1, s2 are two material constants. It is usually assumed that s1 > 0 and s2 ≤ 0, known as E-inequalities, based on the assumption that the free energy function be positive definite for any deformation. In this note, we shall relax this assumption and with a thermodynamic stability analysis, prove that s2 need not be negative so that some typical behavior of materials under contraction can also be modeled.

A study of elasticity and plasticity equations under arbitrary displacements and strains

Equations of geometrically nonlinear theory of elasticity with finite displacements and strains are analyzed. The equations are composed using three versions of physical relations and applied to solve the problem of tension-compression of a straight bar. It is shown that the use of the classical relations between the components of the stress tensor and the Cauchy-Green strain tensor in the problem of compression of the bar results in the appearance of “spurious” static loss of stability such that the bar axis remains straight if the stresses are referred to unit areas before the deformation (conditional stresses). However, in the problem of tension, the classical relations do not permit one to describe the phenomenon of static instability (neck formation as the plastic instability occurs). These drawbacks disappear if one uses the third version of the physical equations, composed as relations between the true stresses referred to unit areas of the deformed faces on which they act and the true elongations and shears. The relations of the third version are most correct; they permit one to pass to self-consistent equations of elasticity and plasticity under small strains and finite displacements, and they should be recommended for practical use. As an example, such relations are composed for the flow theory.

Two formulations of elastoplastic problems and the theoretical determination of the location of neck formation in samples under tension

Two formulations of elastoplastic problems in the mechanics of deformable solids with finite displacements and deformations are investigated. The first of these is formulated starting from the classical geometrically non-linear equations of the theory of elasticity and plasticity, in which the components of the Cauchy–Green strain tensor, associated with the components of the conditional stress tensor by physically non-linear relations according to flow theory in the simplest version of their representation, are taken as a measure of the deformations. The second formulation is based on the introduction of the true tensile and shear strains which, according to Novoshilov, are associated with the components of the true stresses by physical relations of the above-mentioned form. It is shown that, in the second version of the formulation of the problem, the use of the corresponding equations, complied taking account of the elastoplastic properties of the material with correct modelling of the ends of cylindrical samples and the method of loading (stretching) them, enables the location of the formation of a neck to be determined theoretically and enables the initial stage of its formation to be described without making any assumptions regarding the existence of initial irregularities in the geometry of the samples.

A variational theory of hyperbolic Lagrangian Coherent Structures

We develop a mathematical theory that clarifies the relationship between observable Lagrangian Coherent Structures (LCSs) and invariants of the Cauchy–Green strain tensor field. Motivated by physical observations of trajectory patterns, we define hyperbolic LCSs as material surfaces (i.e., codimension-one invariant manifolds in the extended phase space) that extremize an appropriate finite-time normal repulsion or attraction measure over all nearby material surfaces. We also define weak LCSs (WLCSs) as stationary solutions of the above variational problem. Solving these variational problems, we obtain computable sufficient and necessary criteria for WLCSs and LCSs that link them rigorously to the Cauchy–Green strain tensor field. We also prove a condition for the robustness of an LCS under perturbations such as numerical errors or data imperfection. On several examples, we show how these results resolve earlier inconsistencies in the theory of LCS. Finally, we introduce the notion of a Constrained LCS (CLCS) that extremizes normal repulsion or attraction under constraints. This construct allows for the extraction of a unique observed LCS from linear systems, and for the identification of the most influential weak unstable manifold of an unstable node.

Existence theorems in intrinsic nonlinear elasticity

We first show how the displacement-traction problem of nonlinear three-dimensional elasticity can be recast either as a boundary value problem or as a minimization problem over a Banach manifold, where the unknown is the Cauchy–Green strain tensor instead of the deformation as is customary. We then consider the pure displacement problem, and we show that, under appropriate smoothness assumptions on the data, either problem recast in this fashion possesses at least a solution if the applied forces are sufficiently small and the stored energy function satisfies specific hypotheses. In particular, the minimization problem provides an example where the functional is not coercive.

The pure displacement problem in nonlinear three-dimensional elasticity: intrinsic formulation and existence theorems

In this Note, the equations of nonlinear three-dimensional elasticity corresponding to the pure displacement problem are recast either as a boundary value problem, or as a minimization problem, where the unknown is in both cases the Cauchy–Green strain tensor, instead of the deformation as is customary. We then show that either problem possesses a solution if the applied forces are sufficiently small and the stored energy function satisfies specific hypotheses. The second problem provides an example of a minimization problem for a non-coercive functional over a Banach manifold. To cite this article: P.G. Ciarlet, C. Mardare, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Determination and factorization of the Cauchy–Green strain tensors from shears and stretches

In a finite deformation at a particle of a continuous body, we consider any triad of infinitesimal material line elements in the undeformed state. It is shown how the right Cauchy–Green strain tensor C may be determined assuming that the stretches along the edges of the triads are known together with the three shears of the pairs of edges. This leads in a natural way to a factorization of the strain tensor C and a corresponding decomposition of the deformation gradient F . This decomposition generalizes the right “extended polar decomposition” introduced in a previous paper [Ph. Boulanger, M. Hayes, Unsheared triads and extended polar decompositions of the deformation gradient, Int. J. Nonlinear Mech. 36 (2001) 399–420]. Alternatively, we may consider the same triad after deformation. It is shown how the inverse left Cauchy–Green strain tensor B - 1 may be determined assuming that the “resiles” (inverses of the stretches) along the edges of the triads are known together with the three “reverse shears” (opposites of the shears) of the pairs of edges. This leads in a natural way to a factorization of the tensor B - 1 and a decomposition of the inverse deformation gradient F - 1 . The corresponding decomposition of F generalizes the left “extended polar decomposition” of [Ph. Boulanger, M. Hayes, Unsheared triads and extended polar decompositions of the deformation gradient, Int. J. Nonlinear Mech. 36 (2001) 399–420]. The left and right decompositions of F associated with the triads are then related. Particular attention is given to the case of plane strain. In this case the results are made more explicit.

On the dual variable of the Cauchy stress tensor in isotropic finite hyperelasticity

Elastic materials are governed by a constitutive law relating the second Piola–Kirchhoff stress tensor Σ and the right Cauchy–Green strain tensor C = F T F . Isotropic elastic materials are the special cases for which the Cauchy stress tensor σ depends solely on the left Cauchy–Green strain tensor B = F F T . In this Note we revisit the following property of isotropic hyperelastic materials: if the constitutive law relating Σ and C is derivable from a potential ϕ, then σ and ln B are related by a constitutive law derived from the compound potential ϕ ○ exp . We give a new and concise proof which is based on an explicit integral formula expressing the derivative of the exponential of a tensor. To cite this article: C. Vallée et al., C. R. Mecanique 336 (2008).

Robust numerical analysis of homogeneous and non-homogeneous deformations

The present paper is devoted to the modeling of finite deformations of hyperelastic bodies described by Ogden's model. For the finite element implementation, the explicit expressions of the tangent operator and the stress tensor are derived. If the eigenvalues of the right Cauchy–Green strain tensor are distinct, a simple and compact tensor formula, well adapted for numerical implementation, is proposed. A limiting technique is used to take into account the special case of coalescent eigenvalues in which non-differentiability occurs. Three test examples, including homogeneous and non-homogeneous deformations, are proposed to illustrate the developed formulation.

Finite anisotropic elasticity and material frame indifference from a nonequilibrium thermodynamics perspective

A closed set of Eulerian evolution equations for nonisothermal finite isotropic and anisotropic elastic behavior is derived using nonequilibrium thermodynamics. In particular, it is shown that to describe the state of elastic deformation, the deformation gradient F is preferred as internal variable to the more well-known left and right Cauchy-Green strain tensors. With the energy and entropy functions being frame invariant, the stress tensor expression as obtained in terms of F satisfies the principle of material frame indifference. As an alternative to the F -formulation, it is also shown how elasticity can be described in terms of the mass density and the isochoric deformation gradient, F ˜ = F / det ⁡ F 3 , instead.

Modeling cylindrical waves in nonlinear elastic composites

A procedure of deriving nonlinear wave equations that describe the propagation and interaction of hyperelastic cylindrical waves in composite materials modeled by a mixture with two elastic constituents is outlined. Nonlinearity is introduced by metric coefficients, Cauchy-Green strain tensor, and Murnaghan potential. It is the quadratic nonlinearity of all governing relations. For a configuration (state) dependent on the radial coordinate and independent of the angular and axial coordinates, quadratically nonlinear wave equations for stresses are derived and a relationship between the components of the stress tensor and partial strain gradient is established. Four combinations of physical and geometrical nonlinearities in systems of wave equations are examined. Nonlinear wave equations are explicitly written for three of the combinations

Quasistatic anti-plane motion in the simplest theory of nonlinear viscoelasticity

We consider a very simple model in the framework of differential viscoelastic materials which are isotropic and incompressible. In this model the Cauchy stress tensor is split in an elastic part and a dissipative part. The elastic part is derived from a strain-energy density function only of the first invariant of the Cauchy–Green strain tensor. The dissipative part is like the Navier–Stokes equations: linear in the stretching tensor with a constant viscosity parameter. For this model we provide some time and spatial estimates in the quasistatic approximations for the equations governing anti-plane shear motions. Several explicit examples for specific form of the strain energy are produced. Our results impose analytical restrictions on the mathematical properties of the strain energy to ensure a physical behavior in the creep and recovery experiments. Moreover, we show polynomial decay for the spatial behavior in the class of stress-hardening (or strain-stiffening) materials. For stress-softening materials a Phragmen–Lindelof alternative is proved.

A planar biaxial constitutive relation for the luminal layer of intra-luminal thrombus in abdominal aortic aneurysms

The rupture risk of abdominal aortic aneurysms (AAAs) is thought to be associated with increased levels of wall stress. Finite element analysis (FEA) allows the prediction of wall stresses in a patient-specific, non-invasive manner. We have recently shown that it is important to include the intra-luminal thrombus (ILT), present in approximately 70% of AAA, into FEA simulations of AAA. All FEA simulations to date assume an isotropic, homogeneous material behavior for this material. The purpose of this work was to investigate the multi-axial biomechanical behavior of ILT and to derive an appropriate constitutive relation. We performed planar biaxial testing on the luminal layer of nine ILT specimens obtained fresh in the operating room (9 patients, mean age 71±4.5 years, mean diameter 5.9±0.4 cm), and a constitutive relation was derived from this data. Peak stretch and maximum tangential modulus (MTM) values were recorded for the equibiaxial protocol in both the circumferential ( θ) and longitudinal ( L) directions. Stress contour plots were used to investigate the presence of mechanical anisotropy, after which an appropriate strain energy function was fit to each of the specimen datasets. The peak stretch values for the luminal layer of the ILT were (mean±SEM) 1.18±0.02 and 1.13±0.02 in the θ and L directions, respectively ( p = 0.14 ). The MTM values were 20±2 and 23±3 N/cm 2 in the θ and L directions, respectively ( p = 0.37 ). From these results and our observation of the symmetry of the stress contour plots for each specimen, we concluded that the use of an isotropic strain energy function for ILT is appropriate. Each specimen data set was then fit to a second-order polynomial strain energy function of the first invariant of the left Cauchy–Green strain tensor, resulting in an accurate fit (average R 2 = 0.92 ± 0.02 ; range 0.80–0.99). Comparison of our previously reported, uniaxially derived constitutive relation with the biaxially derived relation derived here shows large differences in the predicted mechanical response, underscoring the importance of the appropriate experimental methods used to derive constitutive relations. Further work is merited in an effort to produce more accurate predictions of wall stresses in patient-specific AAA, and viscoelastic behaviors of the ILT.

Quadratically Nonlinear Cylindrical Hyperelastic Waves: Derivation of Wave Equations for Axisymmetric and Other States

A rigorous approach of nonlinear continuum mechanics is used to derive nonlinear wave equations that describe the propagation and interaction of hyperelastic cylindrical waves. Nonlinearity is introduced by means of metric coefficients, the Cauchy—Green strain tensor, and the Murnaghan potential and corresponds to the quadratic nonlinearity of all basic relationships. Quadratically nonlinear wave equations are derived for three states (configurations): (i) axisymmetric configuration dependent on the radial and axial coordinates and independent of the angular coordinate, (ii) configuration dependent on the angular coordinate, and (iii) axisymmetric configuration dependent on the radial coordinate. Four ways of introducing physical and geometrical nonlinearities to the wave equations are analyzed. Six different systems of wave equations are written

Quadratically Nonlinear Cylindrical Hyperelastic Waves: Derivation of Wave Equations for Plane-Strain State

A method is proposed for deriving nonlinear wave equations that describe the propagation and interaction of hyperelastic cylindrical waves. The method is based on a rigorous approach of nonlinear continuum mechanics. Nonlinearity is introduced by means of metric coefficients, Cauchy-Green strain tensor, and Murnaghan potential and corresponds to the quadratic nonlinearity of all basic relationships. For a configuration (state) dependent on the radial and angle coordinates and independent of the axial coordinate, quadratically nonlinear wave equations for stresses are derived and stress-strain relationships are established. Four ways of introducing physical and geometrical nonlinearities to the wave equations are analyzed. For one of the ways, the nonlinear wave equations are written explicitly

Finite amplitude transverse waves in materials with memory

We consider the propagation of finite amplitude plane transverse waves in a class of homogeneous isotropic incompressible viscoelastic solids with memory. It is assumed that the Cauchy stress may be written as the sum of an elastic part and a dissipative viscoelastic part. The elastic part is of the form of the stress corresponding to a Mooney–Rivlin material, whereas the dissipative part depends not only on current but also on previous deformations. The body is first subjected to a homogeneous static deformation. It is seen that two finite amplitude transverse plane waves may propagate in every direction in the deformed body. It is also seen that finite amplitude circularly polarized waves may propagate along either n + or n −, where n +, n − are the normals to the planes of the central circular section of the ellipsoid x · B −1 x = 1. Here B is the left Cauchy–Green strain tensor corresponding to the finite static homogeneous deformation.

Inhomogeneous ?longitudinal? plane waves in a deformed elastic material

By definition, a homogeneous isotropic compressible Hadamard material has the property that an infinitesimal longitudinal homogeneous plane wave may propagate in every direction when the material is maintained in a state of arbitrary finite static homogeneous deformation. Here, as regards the wave, ‘homogeneous’ means that the direction of propagation of the wave is parallel to the direction of eventual attenuation; and ‘longitudinal’ means that the wave is linearly polarized in a direction parallel to the direction of propagation. In other words, the displacement is of the form u = ncos k(n · x − ct), where n is a real vector. It is seen that the Hadamard material is the most general one for which a ‘longitudinal’ inhomogeneous plane wave may also propagate in any direction of a predeformed body. Here, ‘inhomogeneous’ means that the wave is attenuated, in a direction distinct from the direction of propagation; and ‘longitudinal’ means that the wave is elliptically polarized in the plane containing these two directions, and that the ellipse of polarization is similar and similarly situated to the ellipse for which the real and imaginary parts of the complex wave vector are conjugate semi-diameters. In other words, the displacement is of the form u = ℜ{S exp iω(S · x − ct)}, where S is a complex vector (or bivector). Then a Generalized Hadamard material is introduced. It is the most general homogeneous isotropic compressible material which allows the propagation of infinitesimal ‘longitudinal’ inhomogeneous plane circularly polarized waves for all choices of the isotropic directional bivector. Finally, the most general forms of response functions are found for homogeneously deformed isotropic elastic materials in which ‘longitudinal’ inhomogeneous plane waves may propagate with a circular polarization in each of the two planes of central circular section of the \(\mathbb{B}\) n-ellipsoid, where \(\mathbb{B}\) is the left Cauchy-Green strain tensor corresponding to the primary pure homogeneous deformation.

Constructions of unsheared pairs in finite strain

Let P be any point in a continuum subject to deformation. Any plane passing through P contains an infinite number of pairs of infinitesimal material line elements which are unsheared in the deformation. Assuming that the Cauchy–Green strain tensors are known we present a simple geometric construction for the determination of these unsheared pairs whether in the deformed or the undeformed state.