Howard L. Schreyer

A particle method for history-dependent materials

Abstract A broad class of engineering problems including penetration, impact and large rotations of solid bodies causes severe numerical problems. For these problems, the constitutive equations are history dependent so material points must be followed; this is difficult to implement in a Eulerian scheme. On the other hand, purely Lagrangian methods typically result in severe mesh distortion and the consequence is ill conditioning of the element stiffness matrix leading to mesh lockup or entanglement. Remeshing prevents the lockup and tangling but then interpolation must be performed for history dependent variables, a process which can introduce errors. Proposed here is an extension of the particle-in-cell method in which particles are interpreted to be material points that are followed through the complete loading process. A fixed Eulerian grid provides the means for determining a spatial gradient. Because the grid can also be interpreted as an updated Lagrangian frame, the usual convection term in the acceleration associated with Eulerian formulations does not appear. With the use of maps between material points and the grid, the advantages of both Eulerian and Lagrangian schemes are utilized so that mesh tangling is avoided while material variables are tracked through the complete deformation history. Example solutions in two dimensions are given to illustrate the robustness of the proposed convection algorithm and to show that typical elastic behavior can be reproduced. Also, it is shown that impact with no slip is handled without any special algorithm for bodies governed by elasticity and strain hardening plasticity.

Axisymmetric form of the material point method with applications to upsetting and Taylor impact problems

Abstract The material point method is an evolution of particle-in-cell methods which utilize two meshes, one a material or Lagrangian mesh defined over material of the body under consideration, and the second a spatial or Eulerian mesh defined over the computational domain. Although meshes are used, they have none of the negative aspects normally associated with conventional Eulerian or Lagrangian approaches. The advantages of both the Eulerian and Lagrangian methods are achieved by using the appropriate frame for each aspect of the computation, with a mapping between the two meshes that is performed at each step in the loading process. The numerical dissipation normally displayed by an Eulerian method because of advection is avoided by using a Lagrangian step; the mesh distortion associated with the Lagrangian method is prevented by mapping to a user-controlled mesh. Furthermore, explicit material points can be tracked through the process of deformation, thereby alleviating the need to map history variables. As a consequence, problems which have caused severe numerical difficulties with conventional methods are handled fairly routinely. Examples of such problems are the upsetting of billets and the Taylor problem of cylinders impacting a rigid wall. Numerical solutions to these problems are obtained with the material point method and where possible comparisons with experimental data and existing numerical solutions are presented.

Fluid–membrane interaction based on the material point method

The material point method (MPM) uses unconnected, Lagrangian, material points to discretize solids, fluids or membranes. All variables in the solution of the continuum equations are associated with these points; so, for example, they carry mass, velocity, stress and strain. A background Eulerian mesh is used to solve the momentum equation. Data mapped from the material points are used to initialize variables on the background mesh. In the case of multiple materials, the stress from each material contributes to forces at nearby mesh points, so the solution of the momentum equation includes all materials. The mesh solution then updates the material point values. This simple algorithm treats all materials in a uniform way, avoids complicated mesh construction and automatically applies a noslip contact algorithm at no additional cost. Several examples are used to demonstrate the method, including simulation of a pressurized membrane and the impact of a probe with a pre-inflated airbag. Copyright

Using the material‐point method to model sea ice dynamics

[1] The material-point method (MPM) is a numerical method for continuum mechanics that combines the best aspects of Lagrangian and Eulerian discretizations. The material points provide a Lagrangian description of the ice that models convection naturally. Thus properties such as ice thickness and compactness are computed in a Lagrangian frame and do not suffer from errors associated with Eulerian advection schemes, such as artificial diffusion, dispersion, or oscillations near discontinuities. This desirable property is illustrated by solving transport of ice in uniform, rotational and convergent velocity fields. Moreover, the ice geometry is represented by unconnected material points rather than a grid. This representation facilitates modeling the large deformations observed in the Arctic, as well as localized deformation along leads, and admits a sharp representation of the ice edge. MPM also easily allows the use of any ice constitutive model. The versatility of MPM is demonstrated by using two constitutive models for simulations of wind-driven ice. The first model is a standard viscous-plastic model with two thickness categories. The MPM solution to the viscous-plastic model agrees with previously published results using finite elements. The second model is a new elastic-decohesive model that explicitly represents leads. The model includes a mechanism to initiate leads, and to predict their orientation and width. The elastic-decohesion model can provide similar overall deformation as the viscous-plastic model; however, explicit regions of opening and shear are predicted. Furthermore, the efficiency of MPM with the elastic-decohesive model is competitive with the current best methods for sea ice dynamics.

The material point method for simulation of thin membranes

The material-point method (MPM) is extended to handle membranes, which are discretized by a collection of unconnected material points placed along each membrane surface. These points provide a Lagrangian description of the membrane. To solve for the membrane motion, data carried by the material points are transferred to a background mesh where equations of motion are discretized and solved. Then the solution on the background mesh is used to update the membrane material points. This process of combining Lagrangian and Eulerian features is standard in MPM; the modification for membranes involves merely an implementation of the constitutive equation in a local, normal-tangential coordinate system. It is shown that this procedure does, in fact, provide adequate resolution of membranes with thicknesses that can vary substantially from that of the background mesh spacing. A general formulation is given, but the implementation is in a two-dimensional code that provides a proof-of-principle. Numerical examples including a spring, pendulum and a string with initial slack are used to illustrate the method. The string with slack uses an additional modification of the membrane constitutive equation that allows wrinkles to be modeled at low computational cost. Presented also are examples of two disks impacting, pinching a membrane and rebounding, a difficult problem for standard finite element codes. These simulations require a relaxation of the automatic no-slip contact algorithm in MPM. The addition of the capability to model membranes and the new contact algorithm provide a significant improvement over existing methods for handling an important class of problems. Copyright

Arctic Ice Dynamics Joint Experiment (AIDJEX) assumptions revisited and found inadequate

[1] This paper revisits the Arctic Ice Dynamics Joint Experiment (AIDJEX) assumptions about pack ice behavior with an eye to modeling sea ice dynamics. The AIDJEX assumptions were that (1) enough leads were present in a 100 km by 100 km region to make the ice isotropic on that scale; (2) the ice had no tensile strength; and (3) the ice behavior could be approximated by an isotropic yield surface. These assumptions were made during the development of the AIDJEX model in the 1970s, and are now found inadequate. The assumptions were made in part because of insufficient large-scale (10 km) deformation and stress data, and in part because of computer capability limitations. Upon reviewing deformation and stress data, it is clear that a model including deformation on discontinuities and an anisotropic failure surface with tension would better describe the behavior of pack ice. A model based on these assumptions is needed to represent the deformation and stress in pack ice on scales from 10 to 100 km, and would need to explicitly resolve discontinuities. Such a model would require a different class of metrics to validate discontinuities against observations.

Modeling delamination as a strong discontinuity with the material point method

Decohesion is an important failure mode associated with layered composite materials. Here, the energy implications of material softening are explored in a thermodynamic framework with the result that the dissipated energy (fracture energy) is greater than the plastic work of the traction on the failure surface. It is also argued that if the traction and continuum constitutive equations are solved simultaneously, the resulting algorithm is as simple as that for conventional plasticity. For numerical simulations, the material point method displays the attributes of no mesh deformation so that remeshing is not necessary and the continuous tracking of material points avoids the need for remapping history variables such as decohesion. Compatibility is invoked in a weak sense with the result that no special algorithms are needed for mesh realignment along crack surfaces or for double nodes. Example solutions exhibit no sensitivity of delamination propagation with mesh orientation.

ANALYTICAL AND NUMERICAL TESTS FOR LOSS OF MATERIAL STABILITY

Material instability occurs when ellipticity is lost for symmetric constitutive equations. Prior to loss of ellipticity it is possible that the second-order work of Hill or Drucker becomes negative. There are implications in the literature that numerical solutions cease to be meaningful when a material strain softens and the second-order work is not positive. The instant that the second-order work is zero or negative simultaneously with the additional restriction that the strain increments satisfy compatibility is equivalent to the loss of the ellipticity criterion for symmetric constitutive relations. The loss of ellipticity criterion is the appropriate one for identifying when numerical solutions cease to show convergence and may also be a suitable criterion for identifying the instant at which material failure is initiated. An analytical development is provided for loss of ellipticity together with an explicit expression for the normal to the bifurcation plane. Numerical solutions are given for several sample problems. For all cases, the numerical solutions based on the finite element method conform to the theoretical expectations that unique numerical solutions exist prior to the point at which ellipticity is lost.

A Third-Invariant Plasticity Theory for Frictional Materials

ABSTRACT ABSTRACT A nonassociated plasticity theory is given for frictional materials in terms of the first and third invariants of stress and inelastic strain. Features such as strain hardening, strain softening, dilatation, and compactionareexhibited. Limit states and the Ar0-condition are interpreted in a natural manner with the theory. Comparisons of predictions with experimental data for limit states and deformation paths are given for several materials.

Secant structural solution strategies under element constraint for incremental damage

Abstract Two major difficulties in the nonlinear analysis of post-peak structural behavior are the occurrence of an ill-conditioned tangent stiffness matrix around critical points, namely limit and bifurcation points, and the selection of a suitable constrain on the solution path. As an attempt to make failure prediction available in a routine manner, new solution strategies are proposed in which a secant structural stiffness matrix is formulated for incremental damage models, and the solution path is controlled through a suitable measure of failure at the most severely damaged point in the body. Numerical solutions are given for plane strain and plane stress problems to show that snap-back and snap-through associated with shear band formation can be inexpensively predicted.