This stress tensor is the transpose of the nominal stress. But how can i know if s is the cauchy or the 2nd piolakirchhoff stress tensor. Conservation laws and mechanical equilibria continuum. Kirchhoff stress tensor maps normal vectors in the reference configuration to traction vectors defined in the spatial configuration. This is represented by the popular figure to the right showing an. In the absence of an explicit deformation mapping, it is uncommon to give a material description of the atomistic stress tensor. Consider now a vector element of surface in the reference configuration, nds, where ds is the area of the element and n is the unit normal. Why the second piolakirchhoff stress is the derivative of. Kyuichiro washizu on the occasion of his 60th birthday and the completion of a distinguished academic career at the university of tokyo. Continuum mechanics is all about using linear algebra, with some calculus thrown in, to describe the deformations in objects and relate them to the resulting stresses. Here p is the rst piolakirchhoff stress tensor,r x is the 3d nabla operator in.
Recursively, the definition of the tensor s is given in terms of general derivatives of the strain energy functions \partial w \partial c where w is. Dan fleisch briefly explains some vector and tensor concepts from a students guide to vectors and tensors. Chapter 4 discusses a new family of tensors related to the deformation of continuous material. Where c is the 4th order elasticity tensor, and e, v represents youngs modulus and poissons ratio, respectively. Based on the idea of traction we define two new peridynamic stress tensors and which stand, respectively, for analogues of the cauchy and 1st piola. Cauchy and piolakirchhoff versions of the hardy, tsai and viral stresses. A simple lagrangian form of the balance of linear momentum.
Pk stress tensor and the 2nd piolakirchoff pk stress tensor. Topic 3 lagrangian continuum mechanics variables for analysis. Applications are done on one hand by the characterisation of polyurethane 60 65 shore a a compressive material, and on the other hand by the characterisation of polyurethane 95 shore a and fluorosilicone, both incompressive materials. If the constitutive law linking the second piolakirchhoff stress tensor and the right cauchygreen strain tensor derives from a potential, then the cauchy. We show that this effect is also included in the atomistic first piola kirchhoff stress tensor through the motion of the particles. You mentioned that the cauchy stress changes during rigid body rotation and it is not a problem, because this does not increase the stress, it just change the actual. How to express the deformation of an area and volume what are piolakirchhoff and cauchy stresses.
My appreciation for mechanics was nucleated by professors douglas amarasekara and munidasa ranaweera of the then university of ceylon, and was subsequently shaped and grew substantially under the in uence of professors james k. It may be a second piola kirchhoff stress or a logarithmic strain. The terms j and f1 come in to play when you are comparing areas in the reference configuration to those in the current configuration. The most commonly used measure of stress is the cauchy stress tensor, often called simply the stress tensor or true stress. A simple lagrangian form of the balance of linear momentum can be obtained if a from civil 4950 at the hong kong university of science and technology. Cauchy stress tensor kirchhoff stress tensor first piola kirchhoff stress tensor ylhis paper is presented to my good friend, prof. Venant strain tensor and satisfies the principle of material objectivity. It is interesting that most, perhaps even all, stress definitions can be paired with a. Introduction this page introduces several definitions of stress. Besides the cauchy stress tensor often the socalled kirchhoff stress tensor is introduced which results from a push forward of the second piola. The 1st piolakirchhoff stress is the 3d generalization of the 1d concept of engineering stress. The first and second piola kirchhoff stress tensors extend the concept of true and engineering stress to the threedimensional case and operate on area vectors.
We show that the tensor differs from the earlier defined. Kirchhoff stress tensors from classical elasticity. Greennaghdi rate of the kirchhoff stress and deformation. Difference between first and second piolakirchhoff stress vector. In fact, it is formulated as the product of f s where f is the deformation gradient and s is the second piola kirchhoff tensor.
Therefore, we need to develop alternative stress tensors. It is shown that, whenever the residual stress does not vanish, the response function delivering the piola kirchhoff stress in terms of the deformation gradient may be genuinely linear, and yet independent of the observer. The problem with the 2nd piolakirchhoff stress is its physical interpretation. The piolakirchhoff stress may depend linearly on the. This stress, which is internally used in calculix for all applications the socalled total lagrangian approach, see 9, can be transformed into the first piolakirchhoff stress p the socalled engineering stress, a nonsymmetric tensor and. Assessment of second piolakirchhoff stress tensor in. Hardy stress tensor tsai stress tensor virial stress tensor cauchy stress tensor straight bonds noncentral. This permits one to calculate the transverse components of the second piola kirchhoff stress tensor in piezoelectric doublycurved shells with a high.
Using the assumption that the strain and stress states are homogeneous, the integration of 2. This approach is an extension of the piolakirchhoff hybrid stress formulation that we have recently proposed for linear elastostatics 1, and is applied in this communication to. The stress tensor is updated by relating cauchys tensor with the second piolakirchhoff stress tensor. However, the deformed configuration of a body is usually unknown and it is usually what we aim to calculate. We use the generalized neohookean model with dilatational penalty proposed by simo and hughes. Two often used for large deformation mechanics are the 1st piola kirchoff pk stress tensor and the 2nd piola kirchoff pk stress tensor. The sas shell formulation utilizes the objective greenlagrange strain tensor that exactly represents the arbitrarily large rigidbody motions of a shell in any curvilinear coordinate system.
Static fluids support normal stress but will flow under shear stress. This permits one to calculate the transverse components of the second piolakirchhoff stress tensor in piezoelectric doublycurved shells with a high. On the dual variable of the cauchy stress tensor in. First and second piola kirchhoff stresses cauchy stress tensor field is defined over the deformed configuration of a structure body and is directly related to the governing equations of equilibrium and boundary conditions. Four possibilities for stress tensors and their relationship to the cauchy stress. This approach is an extension of the piola kirchhoff hybrid stress formulation that we have recently proposed for linear elastostatics 1, and is applied in this communication to problems in the frequency domain. Up to know, our discussion of continuum mechanics has left out the material itself, instead focusing on balance of forces the produced stress defintions and stress equiblibrium equations, and kinematics, which produced definitions of reference and deformed configurations, displacement, deformation gradient tensor, and small and finite strain tensors. The first piola kirchhoff stress tensor, p relates forces in the. This is not a symmetric tensor and for computational ease, this we use a the second piola kirchoff stress which is symmetric in nature as it is a result of the product between the finger tensor. Made with explain everything for continuum mechanics. Finite deformation and viscoelasticity modeling and test. Piolakirchhoffstress and green lagrange strain tensors to explain some important properties of the 2nd piolakirchhoffstress tensor and the greenlagrange strain tensor, we consider the deformation gradient tensor this tensor captures the straining and the rigid body rotations of the material fibers. This stress, which is internally used in calculix for all applications the socalled total lagrangian approach, see 9, can be transformed into the first piola kirchhoff stress p the socalled engineering stress, a nonsymmetric tensor and. We note that the expression for the first piola kirchhoff stress.
Ftp, 1 where w is the strain energy density per unit of undeformed volume which a priori depends on both f and x, p is the piolakirchhoff stress tensor andt denotes the transposition. Initial configuration an overview sciencedirect topics. Oct, 2015 the answer to this question lies at the realm of intermingling of thermodynamics and nonlinear fem. We introduce a novel hybrid stress finite element formulation for twodimensional linear electrodynamics. A model is considered as a representation of compressive and incompressive elastomeric materials in nonlinear behavior. On the coupling of a commercial finite element package. It is shown that, whenever the residual stress does not vanish, the response function delivering the piolakirchhoff stress in terms of the deformation gradient may be genuinely linear, and yet independent of the observer. New formula for geometric stiffness matrix calculation. In this blog post we will investigate these quantities, discuss why there is a need for so many variations of stresses and strains, and illuminate the consequences for you as a finite element analyst. The cauchy stress tensor defined previously, related area vectors to traction vectors in the current state of deformation of a material object. The vector s 0 is called the nominal stress vector and the tensor t 0 is called the nominal stress tensor.
We show that the tensor differs from the earlier defined peridynamic stress tensor. Chapter 5 then addresses constitutive equations for elastic materials and viscous fluids, which are presented as tensor equations relating the tensor concept of stress to the tensors describing deformation, rate of deformation and rotation. The answer to this question lies at the realm of intermingling of thermodynamics and nonlinear fem. This section uses a significant amount of text from teran et al. Such a definition of a geometric stiffness matrix is independent of the strain tensor chosen. The stress measure consistent with the lagrangian strain is the second piolakirchhoff stress s. Stress tensor symbol areaweighted normal force relations cauchy. The transpose of t 0, namely, t 0 t, is called the first piola kirchhoff stress tensor, after piola 1833 and kirchhoff 1852. The subject of all studies in continuum mechanics, and the domain of all physical quantities, is the material body. Piolakirchhoff stress tensor and the lagrangian cauchy stress tensor. Some such stress measures that are widely used in continuum mechanics, particularly in the computational context, are. Hello, here are 2 pages from my disseration explaining why p the first piolakirchhoff stress has the form that it does. A geometric stress stiffness matrix can be obtained by an equilibrium condition when only the initial stress state and pertinent infinitesimal nodal displacement for each row of the matrix is taken into account. First and second piola kirchhoff stress tensors definitions.
On the dual variable of the cauchy stress tensor in isotropic. The second piolakirchoff stress tensor is the derivative of the helmholtz free energy with respect to the green strain tensor m. The cauchy stress or the 1st pk stress have more straightforward physical interpretation. Results on the interaction between atomistic and continuum models. Continuum mechanics and nonlinear elasticity stefano giordano.
The 1st piolakirchhoff stress is energy conjugate to the deformation gradient. An interesting feature of the atomistic first piolakirchhoff stress tensor is the absence of a kinetic contribution, which in the atomistic cauchy stress tensor accounts for thermal fluctuations. A key discriminator among the different stress tensors is whether they report stress in a materials undeformed, and especially unrotated state, the reference configuration, or in its deformed and rotated state, the current configuration. Mdstresslab computing stress in atomistic simulations. However, several other measures of stress can be defined.
As constitutive theory we adopt a nonlinear hyperelastic material. The first and second piolakirchhoff stress tensors extend the concept of true and engineering stress to the threedimensional case and operate on area vectors. The stress measure consistent with the lagrangian strain is the second piola kirchhoff stress s. Nov 21, 20 in structural mechanics you will come across a plethora of stress and strain definitions. Results on the interaction between atomistic and continuum. We note that the expression for the first piolakirchhoff stress. Samer adeeb first and second piolakirchhoff stress tensors. Considering the relation between interatomic force, f ji f ij, p i can be defined as. Hello everybody, in the results file abaqus gives me the stress tensor s. An interesting feature of the atomistic first piola kirchhoff stress tensor is the absence of a kinetic contribution, which in the atomistic cauchy stress tensor accounts for thermal fluctuations. The 1st pk stress is defined such that the total force resulting from the 1st pk stress multiplied by the normal and. Difference between first and second piolakirchhoff stress.
On the shell theory on the nanoscale with surface stresses holm altenbach, victor eremeyev to cite this version. The first piola kirchoff stress tensor relates the cauchy stress tensor to the. The problem with the 2nd piola kirchhoff stress is its physical interpretation. The elasticity tensor providing the powerconjugation of the greennaghdi rate of the kirchhoff stress and the deformation rate is required, e. First and second piolakirchhoff stress tensors definitions. The transpose of t 0, namely, t 0 t, is called the first piolakirchhoff stress tensor, after piola 1833 and kirchhoff 1852. We show that this effect is also included in the atomistic first piolakirchhoff stress tensor through the motion of. If the material rotates without a change in stress state rigid rotation, the components of the 1st piolakirchhoff stress tensor will vary with material orientation. Basically, p is chosen such that it satisfies eq i. Cauchy stress tensor, first and second piolakirchhoff stress tensor accord ing to the socalled linearized finite theory of elasticity for hyperelastic inter nally. On the shell theory on the nanoscale with surface stresses. The first piola kirchhoff stress tensor p which will be called the pk1 stress for. However, in admal and tadmor 2016a, we have shown that there exists a material description of the atomistic stress in the form of an atomistic piolakirchhoff stress. For large deformations, also called finite deformations, other measures of stress, such as the first and second piola kirchhoff stress tensors, the biot stress tensor, and the kirchhoff stress tensor, are required.
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