Quasi-static Mechanics Research Articles

Collagen V deficiency during murine tendon healing results in distinct healing outcomes based on knockdown severity

Tendon function is dependent on proper organization and maintenance of the collagen I tissue matrix. Collagen V is a critical regulator of collagen I fibrils, and while prior studies have shown a negative impact of collagen V deficiency on tendon healing outcomes, these studies are confounded by collagen V deficiency through tendon development. The specific role of collagen V in regulating healing tendon properties is therefore unknown. By using inducible Col5a1 knockdown models and analyzing gene expression, fibril and histological tendon morphology, and tendon mechanical properties, this study defines the isolated role of collagen V through tendon healing. Patellar tendon injury caused large changes in tendon gene expression, and Col5a1 knockdown resulted in dysregulated expression of several genes through tendon healing. Col5a1 knockdown also impacted collagen fibril size and shape without observable changes in scar tissue formation. Surprisingly, heterozygous Col5a1 knockdown resulted in improved stiffness of healing tendons that was not observed with homozygous Col5a1 knockdown. Together, these results present an unexpected and dynamic role of collagen V deficiency on tendon healing outcomes following injury. This work suggests a model of tendon healing in which quasi-static mechanics may be improved through titration of collagen fibril size and shape with modulation of collagen V expression and activity.

Post-injury tendon mechanics are not affected by tamoxifen treatment

ABSTRACT Purpose A growing interest in the mechanisms that govern tendon healing has resulted in the develop-ment of tools, such as the tamoxifen-inducible mouse knockdown model, to address these questions. However, tamoxifen is a selective estrogen receptor modulator and may interfere with the tendon healing process. The objective of this study was to evaluate the effects of tamoxifen on post-injury tendon mechanics in wild-type mice. Methods The mice underwent treatment at the time of injury using an established mouse injury model and the injured tendons were evaluated 3 weeks post-injury. The treatment contained tamoxifen suspended in corn oil and was compared to a treatment with only corn oil, as well as mice with no treatment. Tendons were evaluated by measuring the quasi-static and viscoelastic mechanics, collagen fiber realignment, cellularity, and nuclear morphology. Results Mechanical testing of the tendons post-injury revealed no changes to viscoelastic mechanics, quasi-static mechanics, or collagen realignment during loading after tamoxifen treatment with the dosage regimen utilized (three daily injections of 4.5 mg/40 g body weight). Additionally, histological analysis revealed no changes to cellularity or cell nuclear shape. Conclusion Overall, this study revealed that tamoxifen treatment at the time of tendon injury did not result in changes to tendon mechanics or the histological parameters at 3 weeks post-injury.

Research on the constitutive relationship of concrete under uniaxial compression in freeze–thaw environment

• The quantitative method was used to analyze the stress–strain curve during freeze–thaw. • The quantitative relationship between freeze–thaw damage parameters and mechanical indices was determined. • The stress–strain curve model of concrete considering freeze–thaw damage variables was established. • The stress–strain curve model reflects the variation law of the curve during freeze–thaw. Through uniaxial compression tests of concrete prism specimens with strength grades ranging from C35 to C65 under without freezing–thawing (F–T) and with rapid F–T cycles, the damage form, stress–strain relationship, quasi-static mechanics performance, and deformation performance of the specimens were obtained. Further, the relationship between the F–T damage variable, and a model for the F–T damage variable stress–strain curves of the concrete were established. After F–T, the concrete prism specimen exhibited a typical split failure with the crack developed along the interface transition zone (ITZ), and severe damage. The F–T cycles reduced the mechanical properties and brittleness, and increased the ductility of concrete. The linear relationship between the peak stress, peak strain, and static elastic modulus of concrete under uniaxial compression and F–T damage was established. Finally, a unified constitutive model of the concrete stress–strain curve relationship considering the F–T damage variable was established. The resulting model can accurately reflect the stress–strain relationship of concrete under uniaxial compression during F–T, which can provide a reference for future research on the constitutive relationship of concrete F–T damage..

On the nonlinear dynamics of a Kresling-pattern origami under harmonic force excitation

Kresling origami-inspired cylindrical structures have received considerable attention due to their effective deployment characteristics, programmable dual-directional and multifunctional properties, along with their ability to support loads. The Kresling origami investigations to date have focused mostly on their geometry, kinematics, and quasi-static mechanics. While recent works have started to uncover the Kresling origami’s transient deployment dynamics and linear vibration/wave propagation features, the rich nonlinear dynamics of Kresling origami-inspired structures under consistent external excitations have yet to be examined and understood. Given that Kresling origami elements are highly nonlinear so their dynamic responses can be extremely complex under external disturbances, and the fact that in many applications they are embedded in mechanical systems that are continuously excited, such as in robotics, air and ground vehicles, and impact, noise, and vibration isolation structures, this is an important and nontrivial research issue that needs to be addressed.To advance the state of the art, the goal of this research is to explore and uncover the nonlinear dynamics of Kresling origami-inspired tubular structures under consistent forced excitations, in particular harmonic loadings. Time and frequency domain analyses are performed and bifurcation diagrams are developed and analyzed for a range of forcing amplitudes and frequencies. This research reveals the complex responses of Kresling origami-inspired systems, i.e., existence of multiple period-1 attractors, period-n (n>1) attractors, jump from one attractor (limit-cycle) to another attractor, and presence of chaotic attractors for a range of input forcing parameters. The noteworthy result of this study also concerns a synchronization of the origami cylinder’s steady-state axial and rotational (twist) displacements. Without such a forced nonlinear dynamic analysis, these findings have not been and cannot be discovered in previous studies. The insights and knowledge uncovered in this investigation are valuable in raising awareness and providing guidance in the design and control of Kresling origami-based structures for realistic engineering operations, where external dynamic disturbances are often unavoidable.

Survival of polymeric microstructures subjected to interrogatory touch.

Polymeric arrays of microrelief structures have a range of potential applications. For example, to influence wettability, to act as biologically inspired adhesives, to resist biofouling, and to play a role in the "feel" of an object during tactile interaction. Here, we investigate the damage to micropillar arrays comprising pillars of different modulus, spacing, diameter, and aspect ratio due to the sliding of a silicone cast of a human finger. The goal is to determine the effect of these parameters on the types of damage observed, including adhesive failure and ploughing of material from the finger onto the array. Our experiments point to four principal conclusions [1]. Aspect ratio is the dominant parameter in determining survivability through its effect on the bending stiffness of micropillars [2]. All else equal, micropillars with larger diameter are less susceptible to breakage and collapse [3]. The spacing of pillars in the array largely determines which type of adhesive failure occurs in non-surviving arrays [4]. Elastic modulus plays an important role in survivability. Clear evidence of elastic recovery was seen in the more flexible polymer and this recovery led to more instances of pristine survivability where the stiffer polymer tended to ablate PDMS. We developed a simple model to describe the observed bending of micropillars, based on the quasi-static mechanics of beam-columns, that indicated they experience forces ranging from 10-4-10-7 N to deflect into adhesive contact. Taken together, results obtained using our framework should inform design considerations for microstructures intended to be handled by human users.

Investigation of the penetration behavior of silicon dioxide epoxy hybrid nanocomposites

The use of nanomaterials is gradually increasing with the progress of nanotechnology. In particular, the production of nanocomposites incorporating nanoparticles is one of the most significant areas in which nanomaterials are being used increasingly. The first objective of this research was to detect the punch shear or penetration resistance behavior and damage mechanisms of hybrid nanocomposites obtained by using silica (SiO2) nanoparticles. For that purpose, six different SiO2 hybrid nanocomposites with different laminations three layer (3La), 5La, 7La, 11La, 15La and 21La and different thicknesses (HC) of 0.95∼4.98 mm, were made by using vacuum assisted transfer molding (VARTM). During this research, quasi-static punch-shear (QS-PS) tests at span punch ratios (SPRs) of 1.16, 1.33, 1.67, 2.00, 2.33, 2.67, and more were conducted to determine quasi-static penetration mechanics and penetration resistance behavior. Moreover, deflection, energy dissipation, damage area, stiffness, and peak force values were investigated through experimental results and scanning electronic microscope (SEM) images.

Effect of post-treatment on the microstructure and mechanical properties of selective laser melted Ti6Al4V lattice structures

Purpose The purpose of this paper is to investigate the post-treatment processes on lattice structures of selective laser melting. Moreover, the effect of pressure during hot isostatic pressing (HIP) is determined. Design/methodology/approach Three post-treatment processes, annealing at 650°C, 920°C and HIP were adopted. The microstructure evolution and mechanical properties of selective lasering melted Ti6Al4V lattice structures after post-treatment were systematically investigated by optical microscope, scanning electron microscope, electron backscattered diffraction, differential scanning calorimetry and quasi-static mechanics tests. Findings The main findings in this paper are as below: first, the pores existing in the samples as-fabricated, annealed at 650°C and 920°C are disappeared after HIP. Second, the microstructure and compressive properties after HIP are similar to that after pure annealing at the same temperature. However, the HIPed sample had the highest number of cycles to failure. Third, the fracture mechanism of as-fabricated samples changes from mixed fracture to the micro-voids accumulation fracture after post-treatment processes. Originality/value HIP post-treatment can be replaced by annealing at the same temperature when the requirement for porosity and fatigue life is not very high.

Dynamics of Kresling origami deployment.

Origami-inspired structures have a rich design space, offering new opportunities for the development of deployable systems that undergo large and complex yet predictable shape transformations. There has been growing interest in such structural systems that can extend uniaxially into tubes and booms. The Kresling origami pattern, which arises from the twist buckling of a thin cylinder and can exhibit multistability, offers great potential for this purpose. However, much remains to be understood regarding the characteristics of Kresling origami deployment. Prior studies have been limited to Kresling structures' kinematics, quasistatic mechanics, or low-amplitude wave responses, while their dynamic behaviors with large shape change during deployment remain unexplored. These dynamics are critical to the system design and control processes, but are complex due to the strong nonlinearity, bistability, and potential for off-axis motions. To advance the state of the art, this research seeks to uncover the deployment dynamics of Kresling structures with various system geometries and operating strategies. A full, six-degrees-of-freedom model is developed and employed to provide insight into the axial and off-axis dynamic responses, revealing that the variation of key geometric parameters may lead to regions with qualitatively distinct mechanical responses. Results illustrate the sensitivity of dynamic deployment to changes in initial condition and small variations in geometric design. Further, analyses show how certain geometries and configurations affect the stiffness of various axial and off-axis deformation modes, offering guidance on the design of systems that deploy effectively while mitigating the effects of off-axis disturbances. Overall, the research outcomes suggest the strong potential of Kresling-based designs for deployable systems with robust and tunable performance.

A new dynamic balancing method of spindle based on the identification energy transfer coefficient

A new vector matching balance method (VMBM) based on the identification of energy transfer coefficient of spindle rotor system (ETCSRS) is proposed in this paper. This method only needs to extract the original vibration signal of the spindle to compute the spindle rotor system unbalanced vector, and the calculation accuracy depends on the identification precision of ETCSRS. The initial parameters of bearings were calculated by applying Newton-Raphson algorithm based on the quasi-static mechanics analysis of rolling bearings. Through establishing the FEA model of spindle, the structural parameters of the rotor shaft system are obtained and modified by modal correction method. First, the principle of VMBM is verified on a Bentley rotor test bench at different rotating speeds. Second, the verification experiment of VMBM on a high-speed spindle is carried out. Furthermore, the VMBM has a better performance than the influence coefficient method (ICM). The method proposed in this paper is very suitable for on-line dynamic balancing, since it greatly improves the balance efficiency.

Identification of Collision Mechanism at Seismogenic Fault Interface Using Finite Element Analysis Involving Plate Bending Applications Using Ant Colony Optimization

AbstractUnderstanding space-time correlation in equilibrium matrix when force of collision involving earthquake stress strain interaction. The proposed study shows that non-linear dynamics of earthquake behaviour simulated with ant colony optimization in short timescale deformation by analysis of the stiffness matrix and the stress strain interaction process of the rock pattern. An improved ant colony optimization combined with local search is proposed for solving this complex optimization problem of finding trigger zones for earthquake occurrences. The disturbances at trigger basins for any system cause the collapse of a subsystem leading to stress evolution and slip due to strain nucleation. The stress strain network based on redistribution of stress accumulation are discretized into four states of low stress and strain and a finite element model is established to identify vertices for the stress-strain component and edges for global coupling effects have been constructed for dynamic monitoring of stress and strain behaviour at triggering zones. In this paper, an efficient algorithm is developed for the formation of null basis of triangular and rectangular plate bending finite element models, corresponding to highly sparse flexibility matrices. Triggering basins serve as harbingers of large earthquake where stress-strain interactions have been analyzed by the quasi-static mechanics of seismic precursory stress-strain propagation in the crustal lithosphere. The simulation framework shows that with time, spatial triggering points as stress varies from one point to another to identify the external influences for the body forces and the surface forces for geodynamic frameworks.

Entropy, Topological Theories and Emergent Quantum Mechanics

The classical thermostatics of equilibrium processes is shown to possess a quantum mechanical dual theory with a finite dimensional Hilbert space of quantum states. Specifically, the kernel of a certain Hamiltonian operator becomes the Hilbert space of quasistatic quantum mechanics. The relation of thermostatics to topological field theory is also discussed in the context of the approach of the emergence of quantum theory, where the concept of entropy plays a key role.

Experimental and simulation study on the thermal characteristics of the high-speed spindle system

High-speed spindles often suffer from degeneration in its machining accuracy caused by the uneven distribution of temperature field. In order to improve the machining accuracy of high-speed spindles, a three-dimensional (3D) finite element analysis (FEA) model, which considered the combined effect of thermal contact resistance (TCR) and the change in heat power and stiffness caused by thermal displacements of bearing components on the accuracy of simulation results, was proposed to conduct transient thermal-structure analysis of high-speed spindles. The predictive model for TCR was proposed based on the fractal theory to characterize the rough surface morphology with disorder, self-affinity and non-stationary random features. And a contact mechanics model was developed to consider the influence of asperities’ deformation on TCR. The thermal-structure model of bearing was proposed to calculate the heat power and stiffness based on the quasi-static mechanics analysis. The FEA model proposed in this paper was used to simulate the temperature field distribution and thermal deformations of the high-speed spindle system. Then thermal characteristic experiments were conducted to validate the effectiveness of this model. The results showed that the FEA model was much more accurate than the traditional model which ignored the above two important factors. The temperature field and thermal errors of the spindle system were analyzed.

Fluid–structure interaction and structural analyses using a comprehensive mitral valve model with 3D chordal structure

Over the years, three-dimensional models of the mitral valve have generally been organized around a simplified anatomy. Leaflets have been typically modeled as membranes, tethered to discrete chordae typically modeled as one-dimensional, non-linear cables. Yet, recent, high-resolution medical images have revealed that there is no clear boundary between the chordae and the leaflets. In fact, the mitral valve has been revealed to be more of a webbed structure whose architecture is continuous with the chordae and their extensions into the leaflets. Such detailed images can serve as the basis of anatomically accurate, subject-specific models, wherein the entire valve is modeled with solid elements that more faithfully represent the chordae, the leaflets, and the transition between the two. These models have the potential to enhance our understanding of mitral valve mechanics and to re-examine the role of the mitral valve chordae, which heretofore have been considered to be 'invisible' to the fluid and to be of secondary importance to the leaflets. However, these new models also require a rethinking of modeling assumptions. In this study, we examine the conventional practice of loading the leaflets only and not the chordae in order to study the structural response of the mitral valve apparatus. Specifically, we demonstrate that fully resolved 3D models of the mitral valve require a fluid-structure interaction analysis to correctly load the valve even in the case of quasi-static mechanics. While a fluid-structure interaction mode is still more computationally expensive than a structural-only model, we also show that advances in GPU computing have made such models tractable. Copyright © 2016 John Wiley & Sons, Ltd.

Simulation and experimental study on the thermally induced deformations of high-speed spindle system

In order to avoid the degeneration of high-speed spindle's machining accuracy in actual machining caused by the uneven distribution of temperature field at the design stage, a three-dimensional (3D) finite element analysis (FEA) model, which considered the combined influence of thermal contact resistance (TCR) and bearing stiffness on the accuracy of simulation results, was proposed to conduct transient thermal-structure interactive analysis of motorized spindles. And the method to calculate the boundary conditions used in the FEM model were discussed in detail, such as the heat loads, convective heat transfer, TCR and bearing stiffness. Based on the quasi-static mechanics analysis of rolling bearing, the transfer relationships among multiple variables in the equilibrium equation set of bearings were analyzed. Newton–Raphson algorithm, which regarded the contact angle as the iteration variable, was proposed to calculate the heat power and stiffness of bearings and improve the convergence of the algorithm, and the termination criteria of contact angle's searching trial calculation was proposed to improve the accuracy of the algorithm. The Weierstrass–Mandelbrot (W–M) function in fractal geometry was used to characterize the rough surface morphology of bearing rings. The fractal parameters were identified by the power spectrum method, and a contact deformation model of asperities was developed to calculate the contact parameters used in the TCR modeling. Then, the predictive model for TCR, which considered the combined effect of the morphology of bearing rings and the contact deformation of asperities, was proposed. Thermal equilibrium experiments were conducted to demonstrate the validity of the model. The results showed that the FEA model can accurately simulate the temperature field and thermal deformation of the spindle system and that the FEA model was much more accurate than the traditional thermal model of the high-speed spindle system which ignored TCR and bearing stiffness.

Thermal characteristics analysis and experimental study on the high-speed spindle system

In order to avoid the sudden failure of high-speed spindles in the actual machining process caused by an excessive temperature rise at the design stage, a three-dimensional (3D) finite element analysis (FEA) model was proposed to conduct transient thermal-structure interactive analysis of a high-speed spindle. The FEA model considered thermal contact resistance (TCR) at solid joints and bearing stiffness to improve the accuracy of traditional thermal models which ignored TCR. However, TCRs at solid joints and bearing stiffness were often ignored in traditional thermal models of high-speed spindles. This caused inaccuracies in traditional thermal models. The heat generation of the built-in motor was calculated based on the efficiency analysis method proposed by Bossmanns and Tu [1]. Based on the quasi-static mechanics analysis of rolling bearing, the heat generation and stiffness of bearings were calculated by applying the Newton-Raphson algorithm to improve the convergence. The Weierstrass-Mandelbrot (W-M) function, a function of fractal parameters, was used to characterize the rough surface morphology of bearing rings. The fractal parameters were identified by the structure function method and the measurement data of bearing ring’s surface morphology, and a contact mechanics model was developed to calculate the contact parameters used in the model of TCR. Then, a new predictive model for TCR was proposed based on M-T model. The above boundary conditions were applied to the FEA model, and thermal equilibrium experiments were conducted to validate the effectiveness of the model. The results showed that the FEA model was much more accurate than the traditional model which ignored TCRs at solid joints and bearing stiffness.

Edelen’s dissipation potentials and the visco-plasticity of particulate media

Following is an elaboration on D. G. B. Edelen’s (1972–1973) nonlinear generalization of the classical Rayleigh-Onsager dissipation potentials and the implications for the models of viscoplasticity. A brief derivation is given via standard vector calculus of Edelen’s potentials and the associated non-dissipative or “gyroscopic” forces and fluxes. It is also shown that certain extensions of Edelen’s formulae can be obtained by means of a recently proposed source-flux relation or “inverse divergence,” a generalization of the classical Gauss-Maxwell construct. The Legendre–Fenchel duality of Edelen’s potentials is explored, with important consequences for rate-independent friction or plasticity. The use of dissipation potentials serves to facilitate the development of viscoplastic constitutive equations, a point illustrated here by the special cases of Stokesian fluid-particle suspensions and granular media. In particular, we consider inhomogeneous systems with particle migration coupled to gradients in particle concentration, strain rate, and fabric. Employing a mixture-theoretic treatment of Stokesian suspensions, one is able to identify particle stress as the work conjugate of the global deformation of the particle phase. However, in contrast to past treatments, this stress is not assumed to be a privileged driving force for particle migration. A comparison is made with models based on extremal dissipation or entropy production. It is shown that such models yield the correct dissipative components of force or flux but generally fail to capture certain non-dissipative, but mechanically relevant components. The significance of Edelen’s gyroscopic forces and their relation to reactive constraints or other reversible couplings is touched upon. When gyroscopic terms are absent, one obtains a class of strongly dissipative or hyperdissipative materials whose quasi-static mechanics are governed by variational principles based on dissipation potential. This provides an interesting analog to elastostatic variational principles based on strain energy for hyperelastic materials and to the associated material instabilities arising from loss of convexity.

An Experimental-Numerical Hybrid Method to Determine Dynamic Elastic-Plastic Fracture Toughness

The dynamic fracture behavior of 7075-T6 aluminum alloy was studied by finite element method to simulate a cracked three-point bending specimen loaded by stress wave loading. In order to determine the elastic-plastic dynamic fracture toughness using quasi-static fracture mechanics theory, the nominal load measured by Hopkinson pressure bar loaded fracture testing system was input into a finite element program to calculate the loading point displacement, and then this displacement was employed to obtain the load-displacement field in the vicinity of the crack tip without the inertia effect, the variation of J-integral as a function of time was established using the load-displacement parameters determined by finite element analysis. The critical J-integral corresponding to crack initiation time detected by a small strain gauge mounted on the three-point bending fracture specimen is determined as an elastic-plastic dynamic fracture toughness (JId). The comparison between the equivalent dynamic fracture toughness(KId) given by the aforementioned procedures and the value measured in previous studies was made to verify the validation of the proposed procedure.

Numerical Studies on Two-Way Coupled Fluid Flow and Geomechanics in Hydrate Deposits

Summary Coupled flow and geomechanics play an important role in the analysis of gas-hydrate reservoirs under production. The stiffness of the rock skeleton and the deformation of the reservoir, as well as porosity and permeability, are directly influenced by (and interrelated with) changes in pressure and temperature and in fluid-(water and gas) and solid-(hydrate and ice) phase saturations. Fluid and solid phases may coexist, which, coupled with steep temperature and pressure gradients, result in strong nonlinearities in the coupled flow and mechanics processes, making the description of system behavior in dissociating hydrate deposits exceptionally complicated. In previous studies, the geological stability of hydrate-bearing sediments was investigated using one-way coupled analysis, in which the changes in fluid properties affect mechanics within the gas-hydrate reservoirs, but with no feedback from geomechanics to fluid flow. In this paper, we develop and test a rigorous twoway coupling between fluid flow and geomechanics, in which the solutions from mechanics are reflected in the solution of the flow problem through the adjustment of affected hydraulic properties. We employ the fixed-stress split method, which results in a convergent sequential implicit scheme. In this study of several hydrate-reservoir cases, we find noticeable differences between the results from one- and two-way couplings. The nature of the elliptic boundary value problem of quasistatic mechanics results in instantaneous compaction or dilation over the domain through loading from reservoir-fluid production. This induces a pressure rise or drop at early times (low-pressure diffusion), and consequently changes the effective stress instantaneously, possibly causing geological instability. Additionally, the pressure and temperature regime affects the various phase saturations, the rock stiffness, porosity, and permeability, thus affecting the fluid-flow regime. These changes are not captured accurately by the simpler one-way coupling. The tightly coupled sequential approach we propose provides a rigorous, twoway coupling model that captures the interrelationship between geomechanical and flow properties and processes, accurately describes the system behavior, and can be readily applied to large-scale problems of hydrate behavior in geologic media.

Comparison of diagonal-implicit, linear-implicit and half-explicit Runge–Kutta methods in non-linear finite element analyses

The method of vertical lines in the case of quasi-static solid mechanics applying constitutive models of evolutionary-type yields after the spatial discretization by means of finite elements a system of differential-algebraic equations. It is of substantial interest how fast, accurate, and stable such computations can be carried out. Moreover, the questions are how simple the implementation can be done and how susceptible a procedure is to programming errors. In this article, this is investigated for half-explicit Runge–Kutta methods that are applied to small and finite strain viscoelasticity. The advantage of the method is given by a non-iterative scheme on element level. Additionally, it turns out that for models where linear elasticity is one ingredient in the constitutive model, the method leads to only one required LU-decomposition at the beginning of the entire computation, and in each time step, only one back-substitution step has to be carried out. This outperforms current finite element computations. Order investigations of various integration schemes and the automatic step-size behavior are studied. This new proposal is compared to a classical Backward-Euler implementation, high-order stiffly accurate diagonally implicit Runge–Kutta, and recently proposed Rosenbrock-type methods. Advantages and disadvantages of the applied schemes are discussed.

Linear complementarity formulation for 3D frictional sliding problems

Frictional sliding on quasi-statically deforming faults and fractures can be modeled efficiently using a linear complementarity formulation. We review the formulation in two dimensions and expand the formulation to three-dimensional problems including problems of orthotropic friction. This formulation accurately reproduces analytical solutions to static Coulomb friction sliding problems. The formulation accounts for opening displacements that can occur near regions of non-planarity even under large confining pressures. Such problems are difficult to solve owing to the coupling of relative displacements and tractions; thus, many geomechanical problems tend to neglect these effects. Simple test cases highlight the importance of including friction and allowing for opening when solving quasi-static fault mechanics models. These results also underscore the importance of considering the effects of non-planarity in modeling processes associated with crustal faulting.