Piola-Kirchhoff Stress Research Articles

Polar Decomposition Theory in Nonlinear Analyses of Solids and Structures

The physical meaning, geometric interpretation, and applications of the polar decomposition theory and Jaumann strains and stresses in nonlinear modeling and analysis of solids and structures are considered. Jaumann strains and stresses prove to be objective geometric measures defined with respect to the deformed system configuration, and constant material stiffnesses obtained from experiments using engineering stresses and strains can be directly used in the constitutive equation. Moreover, the relations of Jaumann strains and stresses to Green-Lagrange strains and Piola-Kirchhoff stresses are derived, and it is shown that Jaumann strains can be easily derived by using a new concept of local displacements.

Hybrid strain based three node flat triangular shell elements—I. Nonlinear theory and incremental formulation

Aspects and theories of nonlinear analysis of structures, with special emphasis on structures that are discretized by the finite element method, are discussed. The updated Lagrangian formulation and the incremental Hellinger-Reissner variational principle are adopted. The independently assumed fields employed are the incremental displacements and incremental strains. Accordingly, the incremental second Piola-Kirchhoff stress and the incremental Washizu strain are selected as the incremental stress and strain measures. Various schemes for the transformation of the second Piola-Kirchhoff stress to Cauchy stress are included. Two versions of linear and nonlinear element stiffness and mass matrices are considered. These are the director and simplified versions. Variable thickness of the shell is considered so as to account for the ‘thinning effect’ due to large strain. Material nonlinearity studied in this paper is of elasto-plastic type with isotropic strain hardening. Cases in which small elastic but large plastic strain condition applies are considered and the J 2 flow theory of plasticity, in conjunction with Ilyushin's yield criterion, is employed. To simplify the derivation of (small displacement) stiffness matrix and to facilitate the derivation of explicit expressions for the element matrices, the non-layered approach has been applied.

Integrated FEM Formulation for Total/Updated-Lagrangian Method in Geometrically Nonlinear Problems

This paper presents a new integrated FEM formulation for geometrically non-linear analyses. There have been two methods to solve the so-called large displacement problem, i.e., the total-Lagrangian method and the updated-Lagrangian method, and two types of FEM programming have been conventionally written. It is shown in the present paper that these two types of programming can be combined by the use of the covariant components of the incremental Green-Lagrange strain tensor and the contravariant components of the second Piola-Kirchhoff stress tensor in the convected coordinate system. The difference is only seen in the transformation of the constitutive tensorial components. The stiffness matrices for the solid and shell elements by this formulation are illustrated in detail, and advantages and disadvantages of the proposed method are also discussed.

Positive definiteness of the elasticity tensor of a residually stressed material

The purpose of this paper is to investigate the positive definiteness of the elasticity tensors associated with linear hyperelastic materials that support a residual stress. The motivation is that positive definiteness of the elasticity tensor is a central issue in establishing uniqueness of solution for boundary value problems. The uniqueness theorems of classical elasticity rely on the physically motivated condition that the work done to deform a body out of the reference configuration should be positive, with the associated result that the classical elasticity tensor is positive definite. For residually stressed bodies, it is not clear whether the condition that the elasticity tensor be positive definite is a reasonable one. It seems physically well motivated that the work done on a residually stressed body to deform it with small strains out of its unloaded configuration should be positive. But it also seems unlikely that the work will be positive pointwise: the residual stress is not uniform, and therefore must take both compressive and tensile values in the body; so it is plausible that, in any particular deformation, there may be points in the body at which the work may be negative if the deformation is such that it relieves the residual stress at the point. Even if the condition that the work be positive in a deformation is accepted, it does not translate directly into the condition that the elasticity tensor be positive definite because, instead of one, there are several elasticity tensors that are commonly used in the description of residually stressed materials, and each of these elasticity tensors is a function of the residual stress. In this paper, work and energy arguments are used to derive conditions on the elasticity tensors that assure that the total work to deform the body will be positive. The relationships of these derived conditions to the positive definiteness of the elasticity tensors are established. By means of an example, it is shown that in some cases the elasticity tensors associated with the first and second Piola-Kirchhoff stresses indeed fail to be positive definite.

A new method for the modeling of geometric nonlinearities in structures

New concepts of local displacements, local engineering stresses and strains, and virtual local rotations are used to derive geometrically exact theories of structures undergoing large rotations and displacements but small strain vibrations. The vector expressions, objectivity, and relationships of several work-conjugate pairs of stress and strain measures are discussed and limitations of their use in the formulation of geometrically nonlinear theories of structures are investigated. The Green-Lagrange strains and second Piola-Kirchhoff stresses are objective (i.e. invariant under rigid-body motions) and complete nonlinear measures, but they are not geometric measures and hence difficult to use. On the other hand, the engineering stresses and strains are easy to use because they are geometric measures and the material stiffnesses obtained from small-strain experiments can be directly used. However, the engineering stresses and strains are not objective measures when finite rotations are involved. To make the engineering measures objective, we employ a coordinate transformation to remove the rigid-body displacements from the total displacements. Then, the remaining, small local displacement field is used to derive objective local engineering strains, which are equivalent to the Jaumann-Biot-Cauchy strains. Moreover, a new interpretation and manipulation of virtual local rotations is introduced, which makes the derivation of geometrically nonlinear structural theories straightforward. A structural formulation using these concepts fully correlates the Newtonian and energy approaches. These new concepts are fully illustrated by deriving a three-dimensional nonlinear Euler-Bernoulli beam theory.

An explicit finite element formulation for very large deformations based on updated material reference frame

An explicit finite element formulation which has the capability of handling very large geometrical changes is developed. The formulation is based on an updated material reference frame and hence a true stress-strain test can be directly applied to properly characterize properties of materials which are subjected to very large deformations. For the large deformations, a consistent formulation based on the continuum mechanics approach is derived. The kinematics is referred to an updated material frame. Body equillibrium is also established based on the updated geometry. The second Piola-Kirchhoff stress tensor and the updated Lagrangian strain tensor are used in the formulation. Numerical examples of frame structures are solved to validate the presented formulation and to demonstrate the importance of imposing material properties properly.

On the irredundant part of the first Piola-Kirchhoff stress tensor

Given is a deformable medium moving and deforming in a (nice) Riemannian manifold N maintaining the shape of a nice compact, bounded manifold M. The physical qualities of this medium are supposed to be characterized by a first Piola-Kirchhoff stress tensor α( j) which is required to depend on the actual configuration j, possibly in a non local way. We determine and investigate via the theory of elliptic operators the irredundant part of α( j). This part yields the same force densities on M as α( j) does. In case N is Euclidean, this irredundant part is the exact part of α( j) extracted via Hodge theory. The medium is equivalently described by α or by a vector field H on the configuration space. Moreover, the virtual work determined by these force densities is studied and discussed in detail. Finally a dynamics is set up within the framework of symplectic geometry in case δM = 0.

Finite strain elastic closed form solutions for some axisymmetric problems

In this paper, rate type constitutive equations using the Jaumann and 2nd Piola-Kirchhoff stress rate are used to develop axisymmetric finite strain, elastic, closed form solutions for a variety of loading conditions. We examined several cases comprizing compression-extension loading conditions, simple shear and cavity expansion conditions. Results from small strain analyses are used to indicate strain ranges for which such analyses will provide satisfactory solutions. For all the cases examined, except simple shear, the 2nd Piola-Kirchhoff stress rate does not appear to be a suitable stress rate to describe a material which follows a rate-type constitutive equation for strains greater than about 40%. The Jaumann stress rate solution shows oscillatory shear stress for axisymmetric simple shear similar to that found earlier by many other authors for rectangular (or cuboidal) condition. Negative excess pore water pressure (suction) at the cavity wall during the expansion of a cylindrical cavity was also observed in Jaumann stress rate solution.

Integrated FEM Formulation for Total/Updated-Lagrangian Method in Geometrically Nonlinear Problems.

This paper presents a new integrated FEM formulation for geometrically nonlinear analyses. There have been two methods to solve the so-called large displacement problem, i.e., the total-Lagrangian method and the updated-Lagrangian method, and two types of FEM programming have been conventionally written. It is shown in the peresent paper that these two types of programming can be combined by the use of the covariant components of the incremental Green-Lagrange strain tensor and the contravariant components of the second Piola-Kirchhoff stress tensor in the convected coordinate system. The difference is only seen in the transformation of the constitutive tensor components. The stiffness matrices for the solid and shell elements by this formulation are illustrated in detail, and advantages and disadvantages of the proposed method are also discussed.

Study on Strength and Nonlinear Stress-Strain Response of Plain Woven Glass Fiber Laminates under Biaxial Loading

Strength and stress-strain properties are investigated for plain woven glass fiber laminates using thin tubular specimens under biaxial static and cyclic loadings. Some composites show large distortion and fiber reorientation close to failure under biaxial load ing. For such composites the usage of the 2nd Piola-Kirchhoff stress instead of the nominal stress is proposed for strength evaluation. The experimental result reveals that the Tsai-Wu criterion fits the biaxial strength expressed by the 2nd Piola-Kirchhoff stress much better than the strength expressed by the nominal stress. Strong interaction between axial and shear deformations is found beyond the knee/yield points while, within the elastic range no interaction is found between the axial and shear stiffness. Fiber misalignment with the loading axis affects the observed stress-strain behavior under cyclic loading. More than two S-S curves are distinguishable when the fiber misalignment is higher than 2°. Such stress-strain relations are successfully simulated using an incremental method in conjunc tion with updating fiber orientation. Good agreement between the calculated result and ex perimental one confirms that the proposed model which considers fiber misalignment can explain the observed stress-strain response under biaxial fatigue loading.

Numerical analysis of sampling disturbances in clay soils

AbstractIn this paper, an insight is provided into the quality of soil samples during the penetration of soil samplers. An updated Lagrangian finite element formulation with the second Piola–Kirchhoff stress rate (the Truesdell stress increment) to account for the large deformation behaviour near the sampling tube is used to determine the mechanical disturbances to a soft clay. The penetration of the sampler is simulated by splitting a group of nodes ahead of the penetration route up to a sufficient depth and applying incremental displacements to match the geometric configuration of the sampling tube. Consolidation effect is included to account for the rate of penetration. Thin‐layer elements are added at the inside wall of the sampling tube to model the soil–sampler interface.The numerical results show that the central core of the sample is subjected to three distinct stages of vertical strain history, compression–extension–recompression, with the primary irrecoverable disturbances due to the compression stage ahead of the sampler. The degree of disturbance for a frictionless sampler was found to be constant after a penetration depth of 75 per cent of the sample tube diameter, while for a frictional sampler the degree of disturbances keeps increasing as the penetration proceeds. The results of a parametric study to determine the influence on sampling disturbances due to the rate of penetration, the thickness and the tip angle of the sample tube and sampler type are also presented.

Adaptive mesh refinement for faceted shells

The application of the Zienkiewicz and Zhu error estimator to three-dimensional faceted shells presents some difficulties which arise from the lack of C1 continuity of the geometry. This paper presents a method to circumvent such difficulties based on the existence of a mapping from a flat reference configuration to the actual shell. This mapping is used not only to enable meshes to be generated in a two-dimensional space, but also to define a stress measure – similar to the first Piola–Kirchhoff stress used in large deformation mechanics – which can be consistently smoothed over the reference configuration. A preliminary investigation of the method is provided by three simple examples.

A nonlinear theory for sandwich shells including the wrinkling phenomenon

This paper presents (on the base of the classical and some additional sandwich assumptions) a geometrically nonlinear theory for sandwich shells with seven kinematic degrees of freedom which is capable to describe the global as well as the local (load singularities, wrinkling) structural behaviour. For the kinematic specification a position vector is used to describe a point of the geometrical middle surface and a corresponding director (no unit vector) as well as a further scalar degree of freedom which could be explained as the linear part of the transversal strain. According to the definition of Green-Lagrangian strains and second Piola-Kirchhoff stress resultants, the nonlinear field equation of force equilibrium with pertinent boundary conditions will be gained from the principal of virtual work. Finally, the equation according to the first order theory for plane structures will be mentioned as a simple special case.

Constitutive laws and force recovery procedures in nonlinear analysis of trusses

Postbuckling analysis of trusses often involves large strain behaviors. For structures with large strains, the use of identical constants in the incremental material laws does not imply identical stress-strain relations, in terms of the 2nd Piola-Kirchhoff stress and Green-Lagrange strain, for formulations with various reference coordinates. Two sets of equations will be derived for recovering the bar forces. One set, referred to as the ‘total form’, is fully nonlinear; and the other, referred to as the ‘incremental form’, is approximate in that the material law is assumed to be piecewise linear. The solutions obtained with the total form are exact and step-size independent, while those obtained with the incremental form are error-prone and step-size dependent for large strain problems.

A 9-node mixed shell element based on the Hu-Washizu principle

This paper proposes a new 9-node degenerated shell element based on a nonlinear mixed formulation. To avoid locking phenomena, we present a mixed formulation based on a three-field Hu-Washizu principle in which displacements, the Green strain tensors, and the second Piola-Kirchhoff stress tensors are independently assumed. In approximating strain and stress fields, covariant components of the strains and stresses measured in the element curvilinear coordinate system are interpolated by the common polynomial functions over an element. Parameter vectors of stress and strain interpolants are elementwise eliminated so that we may obtain an element stiffness matrix similar to that of the displacement model. This formulation is mathematically clear in the variational context, and can include geometrical and material nonlinearities without spoiling such clearness. Numerical results based on our approach are illustrated with satisfactory behavior of the element observed.

On the dead load boundary value problem

The dead load traction boundary condition in finite elasticity is the prescription of the first Piola-Kirchhoff stress vector on the boundary of the undeformed body. Such a prescription does not correspond to an unambiguous physical situation: more than one pair of deformation and actual applied traction on the deformed body can produce the same Piola-Kirchhoff stress vector on the undeformed body. The ambiguous physical meaning of the assignment of the Piola-Kirchhoff traction makes it essential that each solution to a dead load boundary value problem be interpreted in terms of the deformed configuration of the body and the Cauchy traction on the deformed boundary in order to determine the physical situation to which it corresponds. The purpose of this paper is to illustrate the important role played by physical interpretation of solutions through the examination of two familiar problems of dead loading: the biaxial stretching of a sheet of rubber material, studied by Treloar; and the problem, introduced by Rivlin, of a neo-Hookean cube loaded by three equal pairs of forces. In each of these dead load problems the loads are equal but produce multiple deformations most of which are asymmetric. This feature has been thought surprising by a number of authors. Here it is shown that the occurrence of these multiple solutions does not appear unusual when the solutions are interpreted in terms of the physically meaningful Cauchy tractions on the deformed body.

Analysis of viscoelastic behaviour of transversely isotropic materials

AbstractIn this paper a constitutive equation to describe the mechanical behaviour of materials, reinforced with unidirectional fibres, is presented. The material behaviour of both matrix and fibres may be viscoelastic. The constitutive equation is a linear relation between the second Piola–Kirchhoff stress tensor and the Green–Lagrange strain tensor. The effective relaxation functions in the constitutive equation are composed of component relaxation functions employing the structural model of Hashin and Rosen. A two‐dimensional membrane element incorporating this constitutive equation is implemented in a finite element program. The results of several calculations are presented in order to demonstrate the possibilities of the numerical tool. One calculation concerns a square membrane with a circular hole in its centre. The effect of fibre orientation on deformation and stresses will be displayed for this structure as well as for another membrane structure.

A linearized theory of elastic dielectrics which couples electrically to flexural vibrations

Assuming that the free energy depends on the deformation gradient and the spatial electric field, we derive the expressions for the Cauchy stress tensor and the spatial electric displacement from an observer invariant quadratic form of the free energy via the strict definitions of these quantities. Specific forms of the Piola-Kirchhoff stress tensor and the material electric displacement are then deduced and linearized in a particular sense. As an application of the resulting theory, we formulate the problem of an electrically driven disc within the context of the classical bending theory of thin plates. The material of the disc is assumed to have at most the symmetry of a hexagonal system of classC6v.The resulting coupled differential equations for the axial mechanical displacement of the middle surface and the material electric potential indicate that the problem is not empty. This result is of particular interest in view of the fact that it is generally held that the classical theory of piezoelectricity does not permit such couplings to occur.

Mixed variational principles of elastoplasticity in finite deformation.

The present paper proposes mixed variational theorems of classical rate-independent elastoplasticity at finite strain based on the energy equation of Toupin, which takes into account the effects of non-symmetric stress. The proposed variational principle can easily incorporate the internal work related to angular momentum balance, which has not been treated exactly in the previous works. Attention is also focused on systematic and unified derivations for total Lagrangian formulation in terms of the non-symmetric first Piola-Kirchhoff stress as well as the symmetric second Piola-Kirchhoff stress. Also discussed and clarified is the explicit expression of complementary energy in rate type to be applicable for the finite element analysis.

Variational principles for incompressible material and residual work in finite deformation.

The present paper proposes variational principles in finite deformation for a metal alloy showing predominant plastic flow or nearly incompressible material in elastic range, in order to improve the accuracy of numerical analysis. The variational theorems are presented in total Lagrangian formulation in terms of second Piola-Kirchhoff symmetric stress tensor, which can be easily reduced in an updated Lagrangian one. Attention is also focused on the unified derivation of mixed variational principles for residual work related to unbalanced forces which have not been studied previously.