J. C. Simo

STRAIN AND STRESS BASED CONTINUUM DAMAGE MODELS-I. FORMULATION

Abstract Continuum elastoplastic damage models employing irreversible thermodynamics and internal state variables are developed within two alternative dual frameworks. In a strain [stress] -based formulation, damage is characterized through the effective stress [strain] concept together with the hypothesis of strain [stress] equivalence , and plastic flow is introduced by means of an additive split of the stress [strain] tensor . In a strain -based formulation we redefine the equivalent strain , usually defined as the J 2 -norm of the strain tensor, as the (undamaged) energy norm of the strain tensor. In a stress -based approach we employ the complementary energy norm of the stress tensor. These thermodynamically motivated definitions result, for ductile damage, in symmetric elastic-damage moduli. For brittle damage, a simple strain -based anisotropic characterization of damage is proposed that can predict crack development parallel to the axis of loading (splitting mode). The strain- and stress-based frameworks lead to dual but not equivalent formulations, neither physically nor computationally. A viscous regularization of strain-based, rate-independent damage models is also developed, with a structure analogous to viscoplasticity of the Perzyna type, which produces retardation of microcrack growth at higher strain rates. This regularization leads to well-posed initial value problems. Application is made to the cap model with an isotropic strain-based damage mechanism. Comparisons with experimental results and numerical simulations are undertaken in Part II of this work.

A three-dimensional finite-strain rod model. Part II: Computational aspects

Abstract The variational formulation and computational aspects of a three-dimensional finite-strain rod model, considered in Part I, are presented. A particular parametrization is employed that bypasses the singularity typically associated with the use of Euler angles. As in the classical Kirchhoff-Love model, rotations have the standard interpretation of orthogonal, generally noncommutative, transformations. This is in contrast with alternative formulations proposed by Argyris et al. [5–8], based on the notion of semitangential rotation. Emphasis is placed on a geometric approach, which proves essential in the formulation of algorithms. In particular, the configuration update procedure becomes the algorithmic counterpart of the exponential map. The computational implementation relies on the formula for the exponential of a skew-symmetric matrix. Consistent linearization procedures are employed to obtain linearized weak forms of the balance equations. The geometric stiffness then becomes generally nonsymmetric as a result of the non-Euclidean character of the configuration space. However, complete symmetry is recovered at an equilibrium configuration, provided that the loading is conservative. An explicit condition for this to be the case is obtained. Numerical simulations including postbuckling behavior and nonconservative loading are also presented. Details pertaining to the implementation of the present formulation are also discussed.

An analysis of strong discontinuities induced by strain-softening in rate-independent inelastic solids

Ket qualitative features of solutions exhibiting strong discontinuities in rate-independent inelastic solids are identified and exploited in the design of a new class of finite element approximations. The analysis shows that the softening law must be re-interpreted in a distributional sense for the continuum solutions to make mathematical sense and provides a precise physical interpretation to the softening modulus. These results are verified by numerical simulations employing a regularized discontinuous finite element method which circumvent the strong mesh-dependence exhibited by conventional methods, without resorting to viscosity or introducing additional ad-hoc parameters. The analysis is extended to a new class of anisotropic rate-independent damage models for brittle materials.

Algorithms for static and dynamic multiplicative plasticity that preserve the classical return mapping schemes of the infinitesimal theory

A formulation and algorithmic treatment of static and dynamic plasticity at finite strains based on the multiplicative decomposition is presented which inherits all the features of the classical models of infinitesimal plasticity. The key computational implication is this: the closest-point-projection algorithm of any classical simple-surface or multi-surface model of infinitesimal plasticity carries over to the present finite deformation context without modification. In particular, the algorithmic elastoplastic tangent moduli of the infinitesimal theory remain unchanged. For the static problem, the proposed class of algorithms preserve exactly plastic volume changes if the yield criterion is pressure insensitive. For the dynamic problem, a class of time-stepping algorithms is presented which inherits exactly the conservation laws of total linear and angular momentum. The actual performance of the methodology is illustrated in a number of representative large scale static and dynamic simulations.

On a stress resultant geometrically exact shell model. Part II: the linear theory; computational aspects

Abstract Computational aspects of a linear stress resultant (classical) shell theory, obtained by systematic linearization of the geometrically exact nonlinear theory, considered in Part I of this work, are examined in detail. In particular, finite element interpolations for the reference director field and the linearized rotation field are constructed such that the underlying geometric structure of the continuum theory is preserved exactly by the discrete approximation. A discrete canonical, singularity-free mapping between the five and the six degree of freedom formulation is constructed by exploiting the geometric connection between the orthogonal group (SO(3)) and the unit sphere (S 2 ). The proposed numerical treatment of the membrane and bending fields, based on a mixed Hellinger-Reissner formulation,provides excellent results for the 4-node bilinear isoparametric element. As an example, convergent results are obtained for rather coarse meshes in fairly demanding, singularity-dominated, problems such as the classical rhombic plate test. The proposed theory and finite element implementation are evaluated through an extensive set of benchmark problems. The results obtained with the present approach exactly match previous solutions obtained with state-of-the-art implementations based on the so-called degenerated solid approach .

Associated coupled thermoplasticity at finite strains: formulation, numerical analysis and implementation

This paper presents a complete formulation of a model of coupled associative thermoplasticity at finite strains, addresses in detail the numerical analysis aspects involved in its finite element implementation, and assesses the performance of the proposed mechanical and finite element models in a comprehensive set of numerical simulations. On the thermomechanical side, novel aspects of the proposed model of thermoplasticity are (1) the explicit characterization of the plastic (configurational) entropy as an independent internal variable, (2) a thermomechanical extension of the principle of maximum dissipation consistent with the multiplicative decomposition of the deformation gradient, and (3) the exploitation of this extended principle in the formulation of an associative flow which characterizes the evolution of the plastic entropy in terms of the change of the flow criterion with respect to temperature. On the numerical analysis side, salient features of the proposed approach are (4) a new global product formula algorithm constructed via an operator split of the nonlinear initial value problem, which leads to a two-step solution procedure, (5) a unified class of local return mapping algorithms which preserves exactly the incompressibility constraint on the plastic flow and reduces to the classical radial return method for isothermal J2-flow theory, and (6) the formulation of a mixed finite element method in terms of the elastic entropy and the temperature field which circumvents well-known difficulties associated with the incompressibility constraint on the plastic flow. The exact linearization of both the product formula algorithm and an alternative simulataneous solution scheme for the coupled thermomechanical problem is given in two appendices.

On the dynamics of finite-strain rods undergoing large motions a geometrically exact approach

The dynamics of a fully nonlinear rod model, capable of undergoing finite bending, shearing, and extension, is considered in detail. Unlike traditional nonlinear structural dynamics formulations, due to the effect of finite rotations the deformation map takes values in r3 × SO(3), which is a differentiable manifold and not a linear space. An implicit time stepping algorithm that furnishes a canonical extension of the classical Newmark algorithm to the rotation group (SO(3)) is developed. In addition to second-order accuracy, the proposed algorithm reduces exactly to the plane formulation. Moreover, the exact linearization of the algorithm and associated configuration update is obtained in closed form, leading to a configuration-dependent nonsymmetric tangent inertia matrix. As a result, quadratic rate of convergence is attained in a Newton-Raphson iterative solution strategy. The generality of the proposed formulation is demonstrated through several numerical examples that include finite vibration, centrifugal stiffening of a fast rotating beam, dynamic instability and snap-through, and large overall motions of a free-free flexible beam. Complete details on implementation are given in three appendices.

Quasi-incompressible finite elasticity in principal stretches. Continuum basis and numerical algorithms

Abstract The formulation and numerical analysis of constitutive equations for finite elasticity in terms of principal stretches is considered in detail. On the theoretical side, we develop closed-form expressions for the tangent moduli which appear not to have been previously recorded in full generality in the literature. Quasi-incompressible response is accounted for by means of a three-field variational principle which makes use of a multiplicative decomposition of the deformation gradient into volume preserving and dilatational parts. The incompressible limit is enforced by means of a simple and effective augmented Lagrangian procedure. Two possible implementations are discussed. Numerical examples are presented that illustrate the effectiveness of the proposed formulation.

A framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition: part I. continuum formulation

A strain-space formulation of elastoplasticity at finite strains is developed based on a multiplicative decomposition of the deformation gradient. The notion of covariance—which embodies material frame indifference as a particular case—is systematically exploited to uniquely determine reduced forms of the free energy and yield condition that do not preclude anisotropic response. It is shown that the structure of the associative flow rule is uniquely defined as the Kuhn-Tucker optimality condition emanating from the principle of maximum plastic dissipation. Specialization is made to deviatoric plasticity. The isochoric constraint is treated through the exact multiplicative decomposition of the deformation gradient into volume-preserving and spherical parts. As an application, a hyperelastic extension of J2-flow theory is presented with poly-convex hyperelastic uncoupled stored energy function. Computational aspects and large-scale simulations are examined in Part II of this work.

On a stress resultant geometrically exact shell model. Part III: computational aspects of the nonlinear theory

Abstract Computational aspects of a geometrically exact stress resultant model presented in Part I of this work are considered in detail. In particular, by exploiting the underlying geometric structure of the model, a configuration update procedure for the director (rotation) field is developed which is singularity free and exact regardless the magnitude of the director (rotation) increment. Our mixed finite element interpolation for the membrane, shear and bending fields presented in PartII of this work are extended to the finite deformation case. The exact linearization of the discrete form of the equilibrium equations is derived in closed form. The formulation is then illustrated by a comprehensive set of numerical experiments which include bifurcation and post-buckling response, we well as comparisons with closed form solutions and experimental results.