Sensitivity and feasibility of a one-dimensional morphoelastic model for post-burn contraction

We consider a one-dimensional morphoelastic model describing post-burn scar contraction. Contraction can lead to a limited range of motion (contracture). Reported prevalence of burn scar contractures are 58.6% at 3–6 weeks and 20.9% at 12 months post-reconstructive surgery after burns. This model describes the displacement of the dermal layer of the skin and the development of the effective Eulerian strain in the tissue. Besides these components, the model also contains components that play a major role in the skin repair after trauma. These components are signaling molecules, fibroblasts, myofibroblasts, and collagen. We perform a sensitivity analysis for many parameters of the model and use the results for a feasibility study. In this study, we test whether the model is suitable for predicting the extent of contraction in different age groups. To this end, we conduct an extensive literature review to find parameter values. From the sensitivity analysis, we conclude that the most sensitive parameters are the equilibrium collagen concentration in the dermal layer, the apoptosis rate of fibroblasts and myofibroblasts, and the secretion rate of signaling molecules. Further, although we can use the model to simulate significant distinct contraction densities in different age groups, our results differ from what is seen in the clinic. This particularly concerns children and elderly patients. In children we see more intense contractures if the burn injury occurs near a joint, because the growth induces extra forces on the tissue. Elderly patients seem to suffer less from contractures, possibly because of excess skin.

Direct Eulerian formulation of anisotropic hyperelasticity

Abstract Many soft materials and biological tissues comprise isotropic matrix reinforced by fibers in the characteristic directions. Hyperelastic constitutive equations for such materials are usually formulated in terms of a Lagrangean strain tensor referred to the initial configuration and Lagrangean structure tensors defining characteristic directions of anisotropy. Such equations are “pushed forward” to the current configuration. Obtained in this way, Eulerian constitutive equations are often favorable from both theoretical and computational standpoints. Abstract In the present note, we show that the described two-step procedure is not necessary and anisotropic hyperelasticity can be introduced directly in terms of an Eulerian strain tensor and Eulerian structure tensors referring to the current configuration. The newly developed constitutive equation is further applied to the particular case of the transverse isotropy for the sake of illustration.

SHOCK COMPRESSION OF METAL CRYSTALS: A COMPARISON OF EULERIAN AND LAGRANGIAN ELASTIC-PLASTIC THEORIES

An unconventional nonlinear elastic theory is advocated for solids undergoing large compression as may occur in shock loading. This theory incorporates an Eulerian strain measure, in locally unstressed material coordinates. Analytical predictions of this theory and conventional Lagrangian theory for elastic shock stress in anisotropic single crystals of aluminum, copper and magnesium are compared. Eulerian solutions demonstrate greater accuracy compared to atomic simulation (aluminum) and faster convergence with increasing order of elastic constants entering the internal energy. A thermomechanical framework incorporating this Eulerian strain and accounting for elastic and plastic deformations is outlined in parallel with equations for Lagrangian finite strain crystal plasticity. For several symmetric crystal orientations, predicted values of volumetric compression at the Hugoniot elastic limit of the two theories begin to differ substantially when octahedral or prismatic slip system strengths exceed about 1% of the shear modulus. Predicted pressures differ substantially for volumetric compression in excess of 5%. Predictions of Eulerian theory are closer to experimental shock data for aluminum, copper, and magnesium polycrystals.

Alignment of vorticity and rods with Lagrangian fluid stretching in turbulence

Stretching in continuum mechanics is naturally described using the Cauchy–Green strain tensors. These tensors quantify the Lagrangian stretching experienced by a material element, and provide a powerful way to study processes in turbulent fluid flows that involve stretching such as vortex stretching and alignment of anisotropic particles. Analysing data from a simulation of isotropic turbulence, we observe preferential alignment between rods and vorticity. We show that this alignment arises because both of these quantities independently tend to align with the strongest Lagrangian stretching direction, as defined by the maximum eigenvector of the left Cauchy–Green strain tensor. In particular, rods approach almost perfect alignment with the strongest stretching direction. The alignment of vorticity with stretching is weaker, but still much stronger than previously observed alignment of vorticity with the eigenvectors of the Eulerian strain rate tensor. The alignment of strong vorticity is almost the same as that of rods that have experienced the same stretching.

Eulerian elastoplasticity: Basic issues and recent results

Traditional formulations of elastoplasticity in the presence of finite strain and large rotation are Eulerian type and widely used; they are based upon, among other things, the additive decomposition of the stretching or the Eulerian strain-rate into elastic and plastic parts. In such formulations, yield functions and objective rate constitutive equations are expressed in terms of objective Eulerian tensor quantities, including the stretching, the Kirchhoff stress, internal state variables, etc. Each of these quantities transforms in a corotational manner under a change of the observing frame. According to the principle of material frame-indifference or objectivity, each constitutive function should be invariant, whenever the observing frame is changed to another one by any given time-dependent rotation. In this work the general form of constitutive equations is discussed. Several frequently used objective rates are analyzed with respect to their serviceability to develop a self-consistent formulation, i.e. to be integrable to deliver an elastic in particular hyperelastic relation for vanishing plastic deformation. This would be of great importance, e.g., for so-called spring back calculations in metal forming.