Piola-Kirchhoff Stress Research Articles

On the geometry of constraint manifolds

The paper examines the properties that constraint manifolds possess as Riemannian submanifolds of the Euclidean space of all second-order tensors and have implications on the description of mechanical behavior of internally constrained bodies. It is shown that constraint manifolds corresponding to some usual internal constraints have non-zero curvature and, hence, possess a non-Euclidean structure that has to be taken into account when the active and reactive parts of the Piola-Kirchhoff stress are differentiated with respect to deformation gradient.

Modelling the mechanical properties of yeast cells

An analytical model has been developed to describe the compression of a single yeast cell between parallel flat surfaces. Such cells were considered to be thin walled, liquid filled, spheres. Because yeast cells can be compressed at high deformation rates, time dependent effects such as water loss during compression and visco-elasticity of the cell wall could be and were neglected in the model. As in previously published work, a linear elastic constitutive equation was assumed for the material of the cell walls. However, yeast compression to failure requires large deformations, with high wall strains and associated rotations. New model equations appropriate to such high strains with rotations were therefore developed, based on work-conjugate Kirchhoff stresses and Hencky strains. This is an improvement on the earlier use of infinitesimal strains, and on the alternative of Green strains and 2nd Piola–Kirchhoff stresses. It is shown that the choice of stress and strain definition has a significant influence on model predictions for given wall material properties, and will affect estimates of the wall elastic modulus or other wall material property parameters obtained by fitting experimental data.

Drive level dependency in quartz resonators

Common piezoelectric resonators such as quartz resonators have a very high Q and ultra stable resonant frequency. However, due to small material nonlinearities in the quartz crystal, the resonator is drive level dependent, that is, the resonator level of activity and its frequency are dependent on the driving, or excitation, voltage. The size of these resonators will be reduced to one fourth of their current sizes in the next few years, but the electrical power which is applied will not be reduced as much. Hence, the applied power to resonator size ratio will be larger, and the drive level dependency may play a role in the resonator designs. We study this phenomenon using the Lagrangian nonlinear stress equations of motion and Piola–Kirchhoff stress tensor of the second kind. Solutions are obtained using COMSOL for the AT-cut, BT-cut, SC-cut and other doubly rotated cut quartz resonators and the results compared well with experimental data. The phenomenon of the drive level dependence is discussed in terms of the voltage drive, electric field, power density and current density. It is found that the drive level dependency is best described in terms of the power density. Experimental results for the AT-, BT- and SC-cut resonators in comparison with our model results are presented. Results for new doubly rotated cuts are presented. The effects of spurious modes, quality factor and air damping on DLD are presented.

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).

A continuum mechanics based derivation of Reissner’s large-displacement finite-strain beam theory: the case of plane deformations of originally straight Bernoulli–Euler beams

In the present paper, we present a continuum mechanics based derivation of Reissner’s equations for large-displacements and finite-strains of beams, where we restrict ourselves to the case of plane deformations of originally straight Bernoulli–Euler beams. For the latter case of extensible elastica, we succeed in attaching a continuum mechanics meaning to the stress resultants and to all of the generalized strains, which were originally introduced by Reissner at the beam-theory level. Our derivations thus circumvent the problem of needing to determine constitutive relations between stress resultants and generalized strains by physical experiments. Instead, constitutive relations at the stress–strain level can be utilized. Subsequently, this is exemplarily shown for a linear relation between Biot stress and Biot strain, which leads to linear constitutive relations at the beam-theory level, and for a linear relation between the second Piola–Kirchhoff stress and the Green strain, which gives non-linear constitutive relations at the beam theory level. A simple inverse method for analytically constructing solutions of Reissner’s non-linear relations is shortly pointed out in Appendix I.

On the equivalence between spatial and material volume averaging of stress in large strain multi-scale solid constitutive models

This paper discusses some equivalence relationships for large strain multi-scale solid constitutive models based on the volume averaging of the microscopic stress and deformation gradient fields over a representative volume element (RVE). The analysis is carried out within a purely kinematically-based variational framework. Sufficient conditions are established under which the volume average of the microscopic first Piola–Kirchhoff stress over the reference (or material) configuration of the RVE is mechanically equivalent to the average of the microscopic Cauchy stress field over the deformed (or spatial) configuration. The Taylor (or homogeneous deformation) assumption, the affine boundary condition (or linear RVE boundary displacements) as well as the periodic boundary displacement fluctuations constraint – all widely adopted in multi-scale constitutive modelling – satisfy the sufficient conditions. For these classes of models, the choice of the material or spatial configuration for stress volume averaging is immaterial. Some averaging equivalence relations involving the homogenised Kirchhoff stress are also briefly discussed.

Stresses, Singularities, and a Complementary Energy Principle for Large Strain Elasticity

Abstract This article reviews three aspects of large strain elasticity. First, various conjugate stress tensors to strain tensors are reviewed. Many researchers have studied the theory of large strain elasticity. Many stress tensors including the Cauchy stress tensor, the first and the second Piola–Kirchhoff stress tensor, and the Jaumann stress tensor have been proposed to describe the stress state at a point. Recently, the first author of this article proposed a concept of “base forces” to reveal the essence of stress state. By the concept of base forces, the description of the stress state becomes clearer than other stress tensors. We attempt to take base forces as a basic point of view to deal with a review in which different descriptions of stress state are discussed and compared. The governing equations and boundary conditions expressed by the base forces are given. Second, this article reviews the solution of some singularity problems for large strain elasticity, i.e., problems of stress singularity at a crack or a notch tip, at the point of application of a concentrated force and at the vertex of contact in rubberlike materials. Methods of getting the singularity index of stress by using base forces are introduced and compared to earlier work. Complementary energy principles for large strain elasticity have eluded researchers for nearly 100years. A review of some important advances in this is also given, and a new complementary energy principle related to base forces is introduced.

Convergence of Peridynamics to Classical Elasticity Theory

The peridynamic model of solid mechanics is a nonlocal theory containing a length scale. It is based on direct interactions between points in a continuum separated from each other by a finite distance. The maximum interaction distance provides a length scale for the material model. This paper addresses the question of whether the peridynamic model for an elastic material reproduces the classical local model as this length scale goes to zero. We show that if the motion, constitutive model, and any nonhomogeneities are sufficiently smooth, then the peridynamic stress tensor converges in this limit to a Piola-Kirchhoff stress tensor that is a function only of the local deformation gradient tensor, as in the classical theory. This limiting Piola-Kirchhoff stress tensor field is differentiable, and its divergence represents the force density due to internal forces. The limiting, or collapsed, stress-strain model satisfies the conditions in the classical theory for angular momentum balance, isotropy, objectivity, and hyperelasticity, provided the original peridynamic constitutive model satisfies the appropriate conditions.

Postbuckling of composite cylinders under external hydrostatic pressure

Abstract The postbuckling behavior and failure of composite cylinders subjected to external hydrostatic pressure were investigated theoretically using finite element method and experiment. A nonlinear finite element program, ACOSwin, is used for postbuckling progressive failure analyses of composite cylinders. For progressive failure analyses, the complete unloading model associated with the maximum stress criteria is used for post-failure stiffness degradation. In the nonlinear finite element formulation, the updated Lagrangian description method based on the second Piola–Kirchhoff stress tensor and Green strain tensor is used. For the finite element modeling of a composite cylinder, an eight-node shell element is used. A total of five carbon/epoxy composite cylinders were fabricated and tested to verify the finite element results. For comparison, analyses by MSC/NASTRAN and MSC/MARC were additionally conducted. Among the software, the finite element program ACOSwin best predicts the buckling loads within 15% deviation from experimental results for all but one specimen. While the finite element analysis shows global buckling modes with four waves in the hoop direction, the experiment showed only two clear waves.

Two‐point constitutive equations and integration algorithms for isotropic‐hardening rate‐independent elastoplastic materials in large deformation

AbstractThis paper presents alternative forms of hyperelastic–plastic constitutive equations and their integration algorithms for isotropic‐hardening materials at large strain, which are established in two‐point tensor field, namely between the first Piola–Kirchhoff stress tensor and deformation gradient. The eigenvalue problems for symmetric and non‐symmetric tensors are applied to kinematics of multiplicative plasticity, which imply the transformation relationships of eigenvectors in current, intermediate and initial configurations. Based on the principle of plastic maximum dissipation, the two‐point hyperelastic stress–strain relationships and the evolution equations are achieved, in which it is considered that the plastic spin vanishes for isotropic plasticity. On the computational side, the exponential algorithm is used to integrate the plastic evolution equation. The return‐mapping procedure in principal axes, with respect to logarithmic elastic strain, possesses the same structure as infinitesimal deformation theory. Then, the theory of derivatives of non‐symmetric tensor functions is applied to derive the two‐point closed‐form consistent tangent modulus, which is useful for Newton's iterative solution of boundary value problem. Finally, the numerical simulation illustrates the application of the proposed formulations. Copyright © 2008 John Wiley & Sons, Ltd.

A geometrically nonlinear finite element formulation for shells using a particular linearization method

In this paper a particular linearization method is used to derive the updated Lagrangian finite element formulation for geometrically nonlinear analysis of shell structures. Derivation of the formulation is based on rewriting the Green–Lagrange strain and the second Piola–Kirchhoff stress as two second-order functions in terms of a through-the-thickness parameter. Substitution of these two functions into the equilibrium equation, from principle of virtual work, and then using a Taylor series expansion for the nonlinear term lead to a modified linearized incremental equation. In this procedure the stiffness matrices and the internal force vector are consistently derived following the linearization. The results are compared with those of other researchers and good agreements are observed. Advantages of the proposed approach are assessed and comparisons with available solutions are presented.

Dynamic analysis of beam structures considering geometric and constitutive nonlinearity

A fully geometric and constitutive nonlinear model for the description of the dynamic behavior of beam structures is developed. The proposed formulation is based on the geometrically exact formulation for beams due to Simo but, in this article an intermediate curved reference configuration is considered. The resulting deformation map belongs to a nonlinear differential manifold and, therefore, an appropriated version of Newmark’s scheme is used in updating the kinematics variables. Each material point of the cross-section is assumed to be composed of several simple materials with their own constitutive laws. The mixing rule is used to describe the resulting composite. An explicit expression for the objective measure of the strain rate acting on each material point is deduced in this article. Details about its numerical implementation in the time-stepping scheme are also addressed. Viscosity is included at the constitutive level by means of a thermodynamically consistent visco damage model developed in terms of the material description of the First Piola Kirchhoff stress vector. The constitutive part of the tangent tensor is deduced including the effect of rate dependent inelasticity. Finally, several numerical examples, validating the proposed formulation, are given.

On the non-oscillation criterion for multiplicative anisotropic plasticity at large simple shear deformation

The criterion for non-oscillatory stresses under monotonic large simple shear deformation in the context of multiplicative anisotropic plasticity is discussed. In particular, evolving anisotropy combined with a Hill type of yield criterion is considered. It is shown that a sufficient, but not necessary, criterion for a non-oscillatory stress is ellipticity of the first Piola–Kirchhoff stress. Loss of ellipticity corresponds to a critical value h cr of the generalized plastic modulus. Similarly, the absence of limit points on the stress–strain relation motivates an alternative criterion in terms of a critical value h sh ⩽ h cr. Finally, this criterion is demonstrated analytically as well as numerically for an important class of models with evolving anisotropy of the saturation type.

Shear modulus of porcine coronary artery in reference to a new strain measure

To simplify the stress–strain relationship of blood vessels, we define a logarithmic–exponential (log–exp) strain measure to absorb the nonlinearity. As a result, the constitutive relation between the second Piola–Kirchhoff stress and the log–exp strain can be written as a generalized Hooke's law. In this work, the shear modulus of porcine coronary arteries is determined from the experimental data in inflation–stretch–torsion tests. It is found that the shear modulus with respect to the log–exp strain can be viewed as a material constant in the full range of elasticity, and the incremental shear modulus for Cauchy shear stress and small shear strain at various loading levels can be predicted by the proposed Hooke's law. This result further validates the linear constitutive relation for blood vessels when shear deformation is involved.

On the yield functions in Lagrangian stress space of elastic–plastic materials with internal constraints

In the present paper it is shown that the elastic range in the second Piola–Kirchhoff stress space can be chosen in a hyperplane which is through the origin of Lagrangian stress space and perpendicular to the normal of the constraint manifold at the plastic configuration, if the determinate stress response of the elastic–plastic material with simple internal constraints with some condition is correctly chosen, otherwise, it is in general in a hypersurface and the normal flow rule by Il yushin’s postulate will have an indeterminate part. The choice of determinate stress response is probable because of its indeterminacy. Therefore the yield function should be a function of the second Piola–Kirchhoff stress lying in the hyperplane so that it is more simple and the back stress as the geometric center of the elastic range in general is inside the elastic range. Finally some examples are concerned.

Classical results for a non-classical theory: remarks on thermodynamic relations in Green–Naghdi thermo-hyperelasticity

In this contribution we first investigate the thermodynamic’s relation between the entropy flux H and the heat flux Q in the non-classical theory of Green and Naghdi imposed by the entropy equality used in their approach. We show that the classical assumption entropy flux = heat flux/absolute temperature holds for isotropic materials by exploiting the entropy principle of Muller and Liu. As opposed to the usual procedure, we formulate the balance laws with respect to the material description. Subsequently, further restrictions are obtained by exploiting the entropy principle as suggested by Green and Naghdi (in Proc. R. Soc. Lond. 432, 171–194, 1991). This yields the classical potential relations between the entropy and the free energy and between the second Piola-Kirchhoff stress tensor and the free energy.

The mathematical formulation of a generalized Hooke's law for blood vessels

It is well known that the stress–strain relationship of blood vessels is highly nonlinear. To linearize the relationship, the Hencky strain tensor is generalized to a logarithmic–exponential (log–exp) strain tensor to absorb the nonlinearity. A quadratic nominal strain potential is proposed to derive the second Piola–Kirchhoff stresses by differentiating the potential with respect to the log–exp strains. The resulting constitutive equation is a generalized Hooke's law. Ten material constants are needed for the three-dimensional orthotropic model. The nondimensional constant used in the log–exp strain definition is interpreted as a nonlinearity parameter. The other nine constants are the elastic moduli with respect to the log–exp strains. In this paper, the proposed linear stress–strain relation is shown to represent the pseudoelastic Fung model very well.

The gravito-elastodynamics of a pre-stressed elastic earth

SUMMARY The notion that the self-gravitation problem of a perfectly elastic solid body involves small strains from a reference state with arbitrary large initial stresses is reconfirmed and extended. It is shown that the Lagrangian, from which the equations of motion, the boundary conditions and Poisson's equation can be derived in Eulerian or Lagrangian coordinates by the Variational Principle, can be written in a general form which incorporates the various stress measures (Piola-Kirchhoff, Cauchy). Especially for an Eulerian description it is shown that the derived equations of motion and boundary conditions lead to a complete set of mutually orthogonal seismic normal modes. The Lagrangian which Geller (1988) gives is re-evaluated. It appears that in his considerations Geller neglected finite pre-stresses and only accounted for gravitational contributions. This, in general, is not correct. Though the static equilibrium equation has no unique solution for the initial stress components, this does not imply that in physical situations there would not be a specified initial pre-stress. Geller's statement that the asymmetric stress tensor Woodhouse & Dahlen (1978) employ in their Lagrangian necessarily leads to non-conservation of angular momentum is not valid. This stress tensor is a first Piola-Kirchhoff stress tensor, which is a so-called two-point tensor, associating two vector fields defined in different coordinate systems.

Geometrically non‐linear thermoelastic analysis of functionally graded shells using finite element method

AbstractA finite element formulation governing the geometrically non‐linear thermoelastic behaviour of plates and shells made of functionally graded materials is derived in this paper using the updated Lagrangian approach. Derivation of the formulation is based on rewriting the Green–Lagrange strain as well as the 2nd Piola–Kirchhoff stress as two second‐order functions in terms of a through‐the‐thickness parameter. Material properties are assumed to vary through the thickness according to the commonly used power law distribution of the volume fraction of the constituents. Within a non‐linear finite element analysis framework, the main focus of the paper is the proposal of a formulation to account for non‐linear stress distribution in FG plates and shells, particularly, near the inner and outer surfaces for small and large values of the grading index parameter. The non‐linear heat transfer equation is also solved for thermal distribution through the thickness by the Rayleigh–Ritz method. Advantages of the proposed approach are assessed and comparisons with available solutions are presented. Copyright © 2007 John Wiley & Sons, Ltd.

Linear Isotropic Relations in Finite Hyperelasticity: Some General Results

The form of the classical stress–strain relations of linear elasticity are considered here within the context of nonlinear elasticity. For both Cauchy and symmetric Piola-Kirchhoff stresses, conditions are obtained for the associated strain fields so that they are independent of the material constants and compatible with existence of a strain–energy function. These conditions can be integrated in both cases to obtain the most general strain field that satisfies these conditions and the corresponding strain–energy function is obtained. In both cases, a natural choice of form of solution is suggested by the special case of the compatibility conditions being satisfied identically. It will be shown that some strain–energy functions previously introduced in the literature are special cases of the results obtained here. Some recent linear stress–strain relations, proposed in the context of Cauchy elasticity, are examined to see if they are compatible with hyperelasticity.