Piola-Kirchhoff Stress Tensor Research Articles

Projective Peridynamic Modeling of Hyperelastic Membranes With Contact.

Real-time simulation of hyperelastic membranes like cloth still faces a lot of challenges, such as hyperplasticity modeling and contact handling. In this study, we propose projective peridynamics that uses a local-global strategy to enable fast and robust simulation of hyperelastic membranes with contact. In the global step, we propose a semi-implicit strategy to linearize the governing equation for hyperelastic materials that are modeled with peridynamics. By decomposing the first Piola-Kirchhoff stress tensor into a positive and a negative part, successive substitutions can be taken to solve the nonlinear problems. Convergence is guaranteed by further addressing the overshooting problem. Since our global step solve requires no energy summation and dot product operations over the entire problem, it fits into GPU implementation perfectly. In the local step, we further present a GPU-friendly gradient descent method to prevent interpenetration by solving an optimization problem independently. Putting the global and local solves together, experiments show that our method is robust and efficient in simulating complex models of membranes involving hyperelastic materials and contact.

Residual stresses for a new class of transversely isotropic nonlinear elastic solid

We study the response of a class of transversely elastic bodies, wherein the Green–Saint Venant strain tensor is a function of the second Piola–Kirchhoff stress tensor, when the body is residually stressed. The notion of such non-Cauchy elastic bodies being transversely isotropic is defined in Rajagopal (Mech. Res. Commun. 64, 2015, 38–41), and by a body being residually stressed, we mean the interior of the body is not in a stress-free state although the boundary is free of traction as considered by Coleman and Noll (Arch. Ration. Mech. Anal. 15, 1964, 87–111) and by Hoger (Arch. Ration. Mech. Anal. 88, 1985, 271–289).

Three-dimensional nonlinear stress analysis of functionally graded shells subjected to displacement-dependent loads

The nonlinear geometrically exact (GeX) hybrid-mixed four-node solid-shell element under non-conservative loading, developed in the companion paper, is here extended to the stress analysis of functionally graded material (FGM) shells. The term GeX means that the middle surface is described by analytical functions, including spline functions, and, therefore, the coefficients of the first and second fundamental forms can be taken exactly at element nodes. The FGM solid-shell element formulation is based on the choice of N sampling surfaces (SaS) parallel to the middle surface and located at Chebyshev polynomial nodes within the shell as well as on the outer surfaces, to introduce the displacements of these surfaces as basic unknowns. The use of Lagrange polynomials of degree N–1 in the distributions of displacements, strains, stresses, and elastic coefficients through the thickness of the shell leads to an efficient 3N-parameter shell formulation. To develop the FGM four-node solid-shell element based on the hybrid-mixed method, in which strains and stresses are utilized as primary variables, the Hu-Washizu variational principle is applied. Due to the analytical integration used to evaluate the tangent stiffness matrix, the developed FGM solid-shell element subjected to displacement-dependent loads shows superior performance in the case of coarse meshes and allows the use of only one load step in most of the considered benchmark problems. It has been established that the difference between the second Piola-Kirchhoff and Cauchy stress tensors in non-conservative problems for FGM shells undergoing moderately large strains can be significant.

Learning nonlinear constitutive models in finite strain electromechanics with Gaussian process predictors

AbstractThis paper introduces a metamodelling technique that employs gradient-enhanced Gaussian process regression (GPR) to emulate diverse internal energy densities based on the deformation gradient tensor $$\varvec{F}$$ F and electric displacement field $$\varvec{D}_0$$ D 0 . The approach integrates principal invariants as inputs for the surrogate internal energy density, enforcing physical constraints like material frame indifference and symmetry. This technique enables accurate interpolation of energy and its derivatives, including the first Piola-Kirchhoff stress tensor and material electric field. The method ensures stress and electric field-free conditions at the origin, which is challenging with regression-based methods like neural networks. The paper highlights that using invariants of the dual potential of internal energy density, i.e., the free energy density dependent on the material electric field $$\varvec{E}_0$$ E 0 , is inappropriate. The saddle point nature of the latter contrasts with the convexity of the internal energy density, creating challenges for GPR or Gradient Enhanced GPR models using invariants of $$\varvec{F}$$ F and $$\varvec{E}_0$$ E 0 (free energy-based GPR), compared to those involving $$\varvec{F}$$ F and $$\varvec{D}_0$$ D 0 (internal energy-based GPR). Numerical examples within a 3D Finite Element framework assess surrogate model accuracy across challenging scenarios, comparing displacement and stress fields with ground-truth analytical models. Cases include extreme twisting and electrically induced wrinkles, demonstrating practical applicability and robustness of the proposed approach.

Gradient enhanced gaussian process regression for constitutive modelling in finite strain hyperelasticity

This paper introduces a metamodelling technique that leverages gradient-enhanced Gaussian process regression (also known as gradient-enhanced Kriging), effectively emulating the response of diverse hyperelastic strain energy densities. The approach adopted incorporates principal invariants as inputs for the surrogate of the strain energy density. This integration enables the surrogate to inherently enforce fundamental physical constraints, such as material frame indifference and material symmetry, right from the outset. The proposed approach provides accurate interpolation for energy and the first Piola–Kirchhoff stress tensor (e.g. first order derivatives with respect to inputs). The paper presents three notable innovations. Firstly, it introduces the utilisation of Gradient-Enhanced Kriging to approximate a diverse range of phenomenological models, encompassing numerous isotropic hyperelastic strain energies and a transversely isotropic potential. Secondly, this study marks the inaugural application of this technique for approximating the effective response of composite materials. This includes rank-one laminates, for which analytical solutions are feasible. However, it also encompasses more complex composite materials characterised by a Representative Volume Element (RVE) comprising an elastomeric matrix with a centred spherical inclusion. This extension opens the door for future application of this technique to various RVE types, facilitating efficient three-dimensional computational analyses at the macro-scale of such composite materials, significantly reducing computational time compared to FEM2. The third innovation, facilitated by the integration of these surrogate models into a 3D Finite Element computational framework, lies in the assessment of these models scenarios encompassing intricate cases of extreme twisting and more importantly, buckling instabilities in thin-walled structures, thereby highlighting both the practical applicability and robustness of the proposed approach.

Effective Properties of Homogenised Nonlinear Viscoelastic Composites.

We develop a general approach for the computation of the effective properties of nonlinear viscoelastic composites. For this purpose, we employ the asymptotic homogenisation technique to decouple the equilibrium equation into a set of local problems. The theoretical framework is then specialised to the case of a strain energy density of the Saint-Venant type, with the second Piola-Kirchhoff stress tensor also featuring a memory contribution. Within this setting, we frame our mathematical model in the case of infinitesimal displacements and employ the correspondence principle which results from the use of the Laplace transform. In doing this, we obtain the classical cell problems in asymptotic homogenisation theory for linear viscoelastic composites and look for analytical solutions of the associated anti-plane cell problems for fibre-reinforced composites. Finally, we compute the effective coefficients by specifying different types of constitutive laws for the memory terms and compare our results with available data in the scientific literature.

A first-principles method to calculate fourth-order elastic constants of solid materials

A first-principles method is presented to calculate elastic constants up to the fourth order of crystals with the cubic and hexagonal symmetries. The method relies on the numerical differentiation of the second Piola-Kirchhoff stress tensor and a density functional theory approach to calculate the Cauchy stress tensor for a list of deformed configurations of a reference state. The number of strained configurations required to calculate the independent elastic constants of the second, third, and fourth order is 24 and 37 for crystals with the cubic and hexagonal symmetries, respectively. Here, we present conceptual aspects of our method, we provide technical details of its implementation and use, we assess its accuracy, and we discuss several applications. In particular, this method is applied to five crystalline materials with the cubic symmetry (diamond, silicon, aluminum, silver, and gold) and two metals with the hexagonal close packing structure (beryllium and magnesium). Our results are compared to available experimental data and previous computational studies. Calculated linear and nonlinear elastic constants are also used in the context of nonlinear elasticity theory to predict values of volume and bulk modulus over an interval of pressures. These predictions are compared to results obtained from density functional theory calculations to assess the reliability of our method.

On the second Piola-Kirchhoff and Cauchy stress tensors in nonlinear shells subjected to displacement-dependent loads

In this article, a nonlinear geometrically exact (GeX) hybrid-mixed four-node solid-shell element under non-conservative loading using the sampling surfaces (SaS) method is developed. The term “geometrically exact” means that the parametrization of the middle surface is known and, therefore, the coefficients of the first and second fundamental forms and the Christoffel symbols are taken exactly at element nodes. The SaS formulation is based on the choice of N SaS parallel to the middle surface and located at Chebyshev polynomial nodes in order to introduce the displacements of these surfaces as fundamental shell unknowns. Such a choice of unknowns with the use of Lagrange polynomials of degree N–1 in the through-thickness distributions of displacements, strains, and stresses leads to an efficient higher-order shell formulation. To develop the hybrid-mixed four-node solid-shell element, which is free of shear and membrane locking, the Hu–Washizu variational principle is utilized. The proposed GeX hybrid-mixed solid-shell element subjected to follower loads shows superior performance in the case of coarse meshes and allows the use of only one load step in most of the considered numerical examples. It is established that the difference between the second Piola-Kirchhoff and Cauchy stresses in some non-conservative problems for isotropic and composite shells undergoing finite rotations can be significant.

Growth mechanics of the viscoelastic membranes

Living membranes can grow as a response to their environment, and they are also known as viscoelastic materials. However, the growth phenomenon and the viscoelastic behavior of these membranes have been studied separately up to now. The main goal of this paper is to develop a general formulation that can account for growth and viscoelastic behaviors in membranes, simultaneously. Therefore, we propose a formulation for growing quasi-linear viscoelastic membranes. The growth model begins with a multiplicative decomposition of the gradient deformation tensor into elastic and growth tensor, where the growth of the membrane is considered as a single scalar-valued parameter. Furthermore, the quasi-linear viscoelasticity theory is utilized in the elastic configuration as additive decomposition of the second Piola–Kirchhoff stress tensor. To solve the evolution equations for both growth and quasi-linear viscoelastic relations, two implicit second order accuracy integration methods are employed. Moreover, strain-driven and stress-driven growth phenomena are accounted for based on a Total Lagrangian (TL) finite element formulation. The developed formulations can capture growth phenomena and viscoelastic properties of the membranes, individually, and it also enables to model co-existence of growth and viscoelasticity. Eventually, to evaluate the applicability of the developed formulation, several examples are investigated and compared to those available in the literature.

Refined Beam Theory for Geometrically Nonlinear Pre-Twisted Structures

This paper proposes a novel fully nonlinear refined beam element for pre-twisted structures undergoing large deformation and finite untwisting. The present model is constructed in the twisted basis to account for the effects of geometrical nonlinearity and initial twist. Cross-sectional deformation is allowed by introducing Lagrange polynomials in the framework of a Carrera unified formulation. The principle of virtual work is applied to obtain the Green–Lagrange strain tensor and second Piola–Kirchhoff stress tensor. In the nonlinear governing formulation, expressions are given for secant and tangent matrices with linear, nonlinear, and geometrically stiffening contributions. The developed beam model could detect the coupled axial, torsional, and flexure deformations, as well as the local deformations around the point of application of the force. The maximum difference between the present deformation results and those of shell/solid finite element simulations is 6%. Compared to traditional beam theories and finite element models, the proposed method significantly reduces the computational complexity and cost by implementing constant beam elements in the twisted basis.

Continuum damage dynamics of a large-scale flexible multibody system comprised of composite beams

Herein, a continuum damage dynamic model of a large-scale flexible multibody system comprising composite beams is proposed based on the framework of the absolute nodal coordinate formulation. To accurately model the continuum damage dynamics of a multibody system, the Hashin criterion is adopted to describe damage initiation during dynamics. A type of nonlinear evolution law is used to characterize the value of material damage. Furthermore, a material stiffness degradation rule is introduced to describe the process of structural damage. A formulation for the damage element elastic force and its Jacobian are derived based on the second Piola–Kirchhoff stress tensor. Two dynamic numerical examples, including a deployment dynamic analysis of the spatial beam structural unit, are conducted to verify the availability and applicability of the proposed model.

A thermodynamically consistent time integration scheme for non-linear thermo-electro-mechanics

The aim of this paper is the design of a new one-step implicit and thermodynamically consistent Energy–Momentum (EM) preserving time integration scheme for the simulation of thermo-electro-elastic processes undergoing large deformations. The time integration scheme takes advantage of the notion of polyconvexity and of a new tensor cross product algebra. These two ingredients are shown to be crucial for the design of so-called discrete derivatives fundamental for the calculation of the second Piola–Kirchhoff stress tensor, the entropy and the electric field. In particular, the exploitation of polyconvexity and the tensor cross product, enable the derivation of comparatively simple formulas for the discrete derivatives. This is in sharp contrast to much more elaborate discrete derivatives which are one of the main downsides of classical EM time integration schemes. The newly proposed scheme inherits the advantages of EM schemes recently published in the context of thermo-elasticity and electro-mechanics, whilst extending to the more generic case of nonlinear thermo-electro-mechanics. Furthermore, the manuscript delves into suitable convexity/concavity restrictions that thermo-electro-mechanical strain energy functions must comply with in order to yield physically and mathematically admissible solutions. Finally, a series of numerical examples will be presented in order to demonstrate robustness and numerical stability properties of the new EM scheme.

Extreme elastic deformable ceramics on the nanoscale: CrxByOz nanowire as an example

Large quantities of chromium-boron-oxide-based nanowires (NWs) were synthesized via electrodeposition. Transmission electron microscopy investigations led to the conclusion that the deposited oxides formed a homogeneously distributed amorphous structure throughout the whole NW. X-ray photoemission spectroscopy analysis revealed that the prevailing constituents as a statistical ensemble are the (super)hard and brittle CrBO3, Cr2O3, Cr, and B6O phases with the majority of CrBO3. The proposed mathematical model utilizing a non-linear geometrical approximation has been found appropriate to characterize the bending behavior of the NWs without any additional mechanical non-linearity in the small-to-moderate regime. The inhomogeneously evolved strain along the NWs at the highest applied force exhibited an enormous relative displacement of 24% at the most strained region, and the 1st Piola-Kirchhoff stresses have turned out to be in the range of about −25 to +10 GPa. Furthermore, the different contributions in the 1st Piola-Kirchhoff stress tensor components indicate that the presence of a prevailing shear effect can significantly modify the resulting axial stresses. The fitted deflection curves propose varying second-order elastic constants. In general, the determined second-order elastic constants, C 12 and C 44 are slightly lowered with an increasing external applied load. These values are significantly lower than those in earlier experimental and DFT findings for the 2D-, and 3D counterparts of the constituents. The slight reduction with increasing loading is followed by a considerable drop in the second-order elastic constants with 25%–30% at the largest applied concentrated force. The observed reductive figures may stem from structural disordering in the amorphous phase.

Shell finite element formulation for geometrically nonlinear analysis of straight thin-walled pipes

A family of new geometrically nonlinear finite elements is formulated for the simulation of the structural response in the elastic regime of straight pipes with circular cross-sections under various loading conditions. The first Piola–Kirchhoff stress tensor is employed within the principle of virtual work framework in conjunction with a total Lagrangian approach. The Green–Lagrange strain tensor is adopted to capture the finite deformation-small strain effects. The formulations are based on the kinematic assumptions of the thin shell theory and capture the follower effects due to pressure load. Three schemes are proposed to interpolate the displacement fields in the circumferential direction: (1) a Fourier series expansion, (2) a quartic spline interpolation, and (3) a mixed interpolation combining Fourier series and splines. The performance and prediction accuracy of the elements are assessed through comparisons with finite element models based on shell and elbow elements in ABAQUS under various loading conditions. The results demonstrate the ability of the elements to predict the displacement and stress fields. In particular, the element based on Fourier series interpolation is shown to provide accurate predictions.

Pro-Inflammatory Cytokines are Dysregulated in Depression and Correlated With Measures of Psychopathology

This paper presents a kinematic assumption free and thermodynamically consistent non-linear formulation incorporating finite strain and finite deformation for thermoviscoelastic plates/shells based on the conservation and balance laws of the classical continuum mechanics (CCM) in R3 (see Surana and Mathi, (2020) for linear theory). The conservation and balance laws in Lagrangian description with finite strain measure and the conjugate stress measure in R3 are considered. The conjugate pairs in the second law of thermodynamics (SLT), the consideration of additional physics and the principle of equipresence are utilized to determine the constitutive variables and their argument tensors. The constitutive theory for the contravariant second Piola–Kirchhoff stress tensor is derived using Green’s strain tensor and its convected time derivatives of up to order n as its argument tensors using representation theorem with complete basis (i.e. using integrity). The convected time derivatives of the Green’s strain tensor up to order n provide ordered rate constitutive theory for the dissipation mechanism that is naturally non-linear due to Green’s strain measure. The constitutive theory for heat vector derived using representation theorem and integrity is also a non-linear constitutive theory. Simplified linear forms of these constitutive theories are also presented. The solution methods for the mathematical model for the BVPs as well as the IVPs using p-version hierarchical higher degree and higher order finite element method are presented. Due to dissipation, the energy equation is integral part of the mathematical model that accounts for rate of mechanical work resulting in rate of entropy production, hence heat.The plate/shell geometry (flat or curved) is described by its middle surface containing the nodal vectors (at each of the nine nodes), the ends of which define bottom and top surfaces of the plate/shell (Surana and Mathi, 2020). The geometry is mapped into ξηζ natural coordinate space in a two unit cube. The p-version hierarchical local approximations in ξηζ are constructed to describe the deformation of the plate/shell middle surface as well as its faces that is controlled by p-levels in ξ,η and ζ directions (element local approximation). The formulation presented here is accurate for very thin as well as very thick plate/shells and for small as well as finite deformation and finite strains. Non-linear constitutive theories based on integrity allow more complex constitutive behavior in term of strains as well as strain rates, hence dissipation mechanism. Dissipation mechanism is described by an ordered rate theory based on SLT and additional physics. The formulation presented here always ensures thermodynamic equilibrium during the evolution as it is derived using the conservation and balance laws of CCM in R3. The formulation always preserves three dimensional nature of the deformation physics regardless of the plate/shell thickness and is free of locking problems and shear correction factors that plague most of the currently used plate/shell formulations.

Dynamics of pull and release of graphene nanoribbons

The pull and release molecular dynamics simulations of graphene nanoribbons (GNR) lead to their length dependent several folded conformations in the ground states at finite temperatures. The folding occurs when the potential energy of the stretched GNR is sufficient to overcome the energy barriers between the flat GNR and its folded conformations. We explore the dynamics of GNRs during their pull and release as the flat GNR develops intrinsic ripples as soon as thermal fluctuations are introduced and rich mechanical phenomena are found even before the folding process starts. Nudged elastic band calculations were performed to understand the nature of energy barriers and the minimum energy paths between the flat and the rippled nanoribbons. The potential and the kinetic energies of nanoribbons show strong oscillatory characters with attenuation in amplitudes at the start and the end of the pulling process, but as soon as the release process starts the oscillatory nature becomes weak giving rise to small fluctuations. Both thermostatted and unthermostatted nanoribbons were studied with some important observations. Different equivalent continuum models were analyzed which can mimic these experiments. It has been found that the pull and release of the GNRs without any intrinsic rippling can be successfully simulated with a thin plate continuum model using Hooke’s law which does not account for geometric nonlinearities and therefore is aptly able to capture small deformations of the flat GNR. This relation fails when the GNR develops ripples and then undergoes through the pull and release experiments. The rippled GNR was modeled by introducing ripples in the flat continuum GNR by nonlinear curve fitting of the atomistic z-coordinates with a sinusoidal function. Then three different constitutive relations were tried and it was found that modeling GNRs as St. Venant–Kirchhoff materials, which assume a linear relation between the second Piola–Kirchhoff stress tensor and the Green-St. Venant strain tensor correctly reproduce the atomistic behavior before the folding starts. The temperature results were also reproduced by solving the energy balance equation where a thermoelastic constitutive relation between the second Piola–Kirchhoff stress tensor and the Green-St. Venant strain tensor subtracted by the thermal strain tensor was assumed.

Elasticity theory in general relativity

The general relativistic theory of elasticity is reviewed from a Lagrangian, as opposed to Eulerian, perspective. The equations of motion and stress–energy–momentum tensor for a hyperelastic body are derived from the gauge–invariant action principle first considered by DeWitt. This action is a natural extension of the action for a single relativistic particle. The central object in the Lagrangian treatment is the Landau–Lifshitz radar metric, which is the relativistic version of the right Cauchy–Green deformation tensor. We also introduce relativistic definitions of the deformation gradient, Green strain, and first and second Piola–Kirchhoff stress tensors. A gauge-fixed description of relativistic hyperelasticity is also presented, and the nonrelativistic theory is derived in the limit as the speed of light becomes infinite.

Construction of PZT-5H mechano-electric model based on strain rate dependence and its numerical simulation in overload igniter application

Although the force-electricity relationship of piezoelectric ceramics under pulse stress is found to be nonlinear, there is still lacking a reliable model describing this relationship. To bridge this research gap, the mechanical and electrical responses of PZT-5H (Pb1·0[Zr0·49Ti0·46(Nb0·25Sb0.75)0.05]1.0O3) under different strain rates were investigated through uniaxial compression tests and split-Hopkinson pressure bar (SHPB) experiments with an additional electrical output measurement system. With reference to the nonlinear viscoelastic constitutive equation and the piezoelectric equation, a PZT-5H mechano-electric model considering the strain rate effects was built based on the experimental data. The incremental form of the mechano-electric model was established based on the Piola-Kirchhoff stress tensor and Green strain tensor theory. The model was numerically simulated with a user-defined material subroutine in the explicit finite element software ABAQUS. The accuracy of the material subroutine was verified with finite element models of the uniaxial quasi-static compression and the high strain rate SHPB tests. The finite element simulation results were verified with the experimental results, and the two agreed well. The material subroutine was then adopted to predict the mechanical and electrical response of the piezoelectric ceramics inside the overload igniter during projectile launch and impact. The analysis results show that the overload igniter can be stably activated within 200 μs under the launch overload and successfully ignited within 5 μs at speed as low as 50 m/s under the impact overload.

Finite element solver for data-driven finite strain elasticity

A nominal finite element solver is proposed for data-driven finite strain elasticity. It bypasses the need for a constitutive model by considering a database of deformation gradient/first Piola–Kirchhoff stress tensors pairs. The boundary value problem is reformulated as the constrained minimization problem of the distance between (i) the mechanical states, i.e. strain–stress, in the body and (ii) the material states coming from the database. The corresponding constraints are of two types: kinematical, i.e. displacement–strain relation, and mechanical, i.e. conservation linear and angular momenta. The solver uses alternated minimization: the material states are determined from a local search in the database using an efficient tree-based nearest neighbor search algorithm, and the mechanical states result from a standard constrained minimization addressed with an augmented Lagrangian approach. The performance of the solver is demonstrated by means of 2D sanity check examples: the data-driven solution converges to the classical finite element solution when the material database increasingly approximates the constitutive model. In addition, we demonstrate that the balance of angular momentum, which was classically not taken into account in previous data-driven studies, must be enforced as a constraint to ensure the convergence of the method.

A class of transversely isotropic non-linear elastic bodies that is not Green elastic

An implicit constitutive relation is proposed to study transversely isotropic bodies. The relation is obtained assuming the existence of a Gibbs potential that depends on the second Piola–Kirchhoff stress tensor, from which the Green Saint-Venant strain tensor is obtained as the derivative with respect to the stress. The responses of unconstrained as well as inextensible bodies are studied, and some boundary value problems are analysed. An inextensible body, where the constraint of inextensibility appears only in tension is also considered.