Nonuniform Stress Field Determination Based on Deformation Measurement

We demonstrate a technique that, under certain circumstances, will determine stresses associated with a nonuniform deformation field without knowing the detailed constitutive behavior of the deforming material. This technique is based on (1) a detailed deformation measurement of a domain and (2) the observation that for isotropic materials, the strain and the stress, which form the so-called work-conjugate pair, are co-axial, or their eigenvectors share the same direction. The particular measures for strain and stress considered are the Lagrangian strain and the second Piola-Kirchhoff stress. The deformation measurement provides the field of the principal stretch orientation θλ and since the Lagrangian strain and the second Piola-Kirchhoff stress are co-axial, the principal stress orientation θs of the second Piola-Kirchhoff stress is determined. The Cauchy stress is related to the second Piola-Kirchhoff stress through the deformation gradient tensor, which can be measured experimentally. We then show that the principal stress orientation θσ of the Cauchy stress is the sum of the principal stretch orientation θλ and the local rigid-body rotation θq, which is determinable by the deformation gradient through polar decomposition. With the principal stress orientation θσ known, the equation of equilibrium, now in terms of the two principal stresses σ1 and σ2, and θσ, can be solved numerically with appropriate traction boundary conditions. The technique is then applied to the experimental case of nonuniform deformation of a PVC sheet with a circular hole and subject to tension. Limitations and restrictions of the technique and possible extensions will be discussed.

Consistent numerical implementation of hypoelastic constitutive models

In hypoelastic constitutive models, an objective stress rate is related to the rate of deformation through an elasticity tensor. The Truesdell, Jaumann, and Green–Naghdi rates of the Cauchy and Kirchhoff stress tensors are examples of the objective stress rates. The finite element analysis software ABAQUS uses a co-rotational frame which is based on the Jaumann rate for solid elements and on the Green–Naghdi rate for shell and membrane elements. The user subroutine UMAT is the platform to implement a general constitutive model into ABAQUS, but, in order to update the Jacobian matrix in UMAT, the model must be expressed in terms of the Jaumann rate of the Kirchhoff stress tensor. This study aims to formulate and implement various hypoelastic constitutive models into the ABAQUS UMAT subroutine. The developed UMAT subroutine codes are validated using available solutions, and the consequence of using wrong Jacobian matrices is elucidated. The UMAT subroutine codes are provided in the “Electronic Supplementary Material” repository for the user’s consideration.

Equilibrated stress space for nonlinear dimensionally reduced shell models in terms of first-order stress functions

Considering the power series expansion of the three-dimensional variables with respect to the shell thickness coordinate, an equilibrated stress space for the first Piola-Kirchhoff stress vectors is derived in convective curvilinear coordinate system. The infinite series of the two-dimensional translational equilibrium equations are satisfied by introducing two first-order stress function vectors expanded into power series. For the important case of thin shells, the infinite number of two-dimensional equilibrium equations is truncated to obtain a ‘first-order’ model, where the equilibrated stress-space requires three vectorial stress function coefficients only. The formulation presented for thin shells is compared to the nonlinear equilibrium equations of the classical shell theories, written in terms of the first Piola-Kirchhoff stress resultants and stress couples and satisfied by the introduction of three first-order stress function vectors.

Correction to: An efficient and stabilised SPH method for large strain metal plastic deformations

The symbol was introduced incorrectly inside the “Time-stepping the solution” box, directly under the “Compute first Piola–Kirchhoff stress tensor Pi” as in “Appendix A” listing.