阅读说明：本技术 仿真工具 (Simulation tool ) 是由 G·A·阿佐曼迪斯 于 2019-12-09 设计创作，主要内容包括：一种用于仿真针对材料的应变诱发正交各向异性的存储在非暂时性介质上并由处理器执行的方法包括：计算仿真材料的三(3)个主应变方向；计算针对仿真材料的三(3)个畸变应变；以及计算针对仿真材料的三(3)个膨胀应变。该方法还包括计算针对仿真材料的自由能,计算的自由能是根据仿真材料的计算的三个主方向、三个畸变应变和三个膨胀应变来计算的。该方法还进一步包括经由计算的自由能来计算基于针对仿真材料的计算的自由能的针对仿真材料的应力。(A method stored on a non-transitory medium and executed by a processor for simulating strain-induced orthogonal anisotropy for a material, comprising: calculating three (3) main strain directions of the simulation material; calculating three (3) distortion strains for the simulation material; and calculating three (3) expansion strains for the simulated material. The method also includes calculating a free energy for the simulated material, the calculated free energy being calculated from the calculated three principal directions, the three distortion strains and the three expansion strains of the simulated material. The method still further includes calculating a stress for the simulation material based on the calculated free energy for the simulation material via the calculated free energy.)

1. A method stored on a non-transitory medium and executed by a processor for simulating strain-induced orthogonal anisotropy for a material, the method comprising:

calculating three (3) main strain directions of the simulation material;

calculating three (3) distortion strains for the simulation material;

calculating three (3) expansion strains for the simulation material;

calculating a free energy for the simulated material, the calculated free energy being calculated from the three principal directions, the three distortion strains and the three expansion strains calculated for the simulated material; and

calculating, via the calculated free energy, a stress for the simulation material based on the calculated free energy for the simulation material.

2. The method of claim 1, wherein the expansion energy is defined from a large strain point of view according to the following equation:

where z is the expansion function and ε is in the main strain direction.

3. The method of claim 1, further comprising:

defining the distortion strain for a face as the logarithm of the stretch ratio of the simulated material according to the equation:

whereinEqual to pure shear at small strains,andis the true strain in the main direction,andis a stretching in a vertical direction along the simulation material; and

defining the distortion strain for the remaining surface as the logarithm of the stretch ratio of the simulated material according to the following equation:

wherein。

4. The method of claim 1, wherein the calculated stress is calculated in a principal orthogonal anisotropy direction according to the following equation:

。

5. the method of claim 1, further comprising computing entropy elasticity using a cross-linked network in parallel with a generalized maxwell model, the maxwell element comprising a nonlinear spring storing energy as a volume to gibbs free energy, wherein stress is derived according to the following equation:

。

6. a method stored on a non-transitory medium and executed by a processor for simulating stress and strain for an orthotropic composite material, the method comprising:

calculating six (6) distortion strains for the simulated orthotropic composite material;

calculating three (3) expansion strains for the simulated orthotropic composite;

calculating a free energy for the simulated orthotropic composite, the calculated expansion energy being calculated from the six distortion strains and the three expansion strains; and

calculating a stress for the simulated orthotropic composite material based on the calculated expansion energy for the orthotropic material via the calculated free energy.

7. The method of claim 6, wherein the expansion energy is defined from a large strain perspective according to the following equation:

where ε is the strain in the main direction of the orthogonal anisotropy, κ is the bulk modulus, and the z-function combines the expansion contribution to free energy.

8. The method of claim 6, wherein the distortion strain is defined by an angle that results in a hyperbolic secant function in a stress tensor calculation.

9. The method of claim 6, further comprising defining the distortion strain as a logarithm of the stretch ratio of the simulated material according to the equation:

whereinEqual to pure shear at small strains,andis the true strain in the main direction,andis a stretching in a vertical direction along the simulation material; and

defining the distortion strain for the residual surface as a logarithm of the stretch ratio of the simulated material according to the equation:

wherein。

10. The method of claim 6, further comprising computing entropy elasticity using a cross-linked network in parallel with a generalized Maxwell model, the Maxwell elements comprising nonlinear springs storing energy as a volume to Gibbs free energy, wherein stress is derived from the equation:

。

11. the method of claim 6, wherein the calculated stress is calculated in a principal orthogonal anisotropy direction according to the following equation:

。

Technical Field

The present disclosure relates generally to simulation (or simulation), and more particularly to viscoelastic and engineering simulations.

Background

As early as hundreds of years before engineering existed as a subject, mathematicians opened the field of mechanics. The frames they developed to link stress to strain are mathematically feasible but have little engineering intuition. Thus, complex interdisciplinary problems are impractical, or even impossible to solve.

Disclosure of Invention

The present disclosure relates to a method stored on a non-transitory medium and executed by a processor for simulating strain-induced orthogonal anisotropy for a material, the method comprising: calculating three (3) main strain directions of the simulation material; calculating three (3) distortion strains for the simulation material; and calculating three (3) expansion strains for the simulated material. The method also includes calculating a free energy for the simulated material, the calculated free energy being calculated from the calculated three principal directions, the three distortion strains and the three expansion strains of the simulated material. The method still further includes calculating a stress for the simulation material based on the calculated free energy for the simulation material via the calculated free energy.

In some configurations, the expansion can be defined from the perspective of large strain according to the following equation:

where z is the expansion function and ε is in the main strain direction.

In some configurations, the method further includes defining the distortion strain as a logarithm of the stretch ratio of the simulated material according to the equation:

wherein, in the step (A),equal to pure shear at small strains,andis the true strain in the main direction,andis the stretch in the vertical direction along the simulated material and the distortion strain of the remaining surface is defined as the logarithm of the stretch ratio of the simulated material according to the following equation:

wherein。

In some configurations, the method further comprises calculating a calculated stress in the principal strain direction according to the following equation:

。

in some configurations, the method further comprises calculating entropy elasticity using a cross-linked network in parallel with a generalized maxwell model, the maxwell element comprising a nonlinear spring storing energy as a volume ratio gibbs free energy, wherein the stress is derived according to the following equation:

。

the present disclosure also relates to a method stored on a non-transitory medium and executed by a processor for simulating stress and strain for an orthotropic composite, the method comprising calculating six (6) distortion strains for the simulated orthotropic composite and calculating three (3) expansion strains for the simulated orthotropic composite. The method also includes calculating a free energy for the simulated orthotropic composite, the calculated expansion energy being calculated from six distortion strains and three expansion strains. The method still further includes calculating a stress for the simulated orthotropic composite material based on the calculated expansion energy for the orthotropic material via the calculated free energy.

In some configurations, the expansion can be defined from the perspective of large strain according to the following equation:

where ε is the strain in the main direction of the orthogonal anisotropy, κ is the bulk modulus, and the z-function combines the expansion contribution to the free energy.

In some configurations, the distortion strain is defined by an angle that results in a hyperbolic secant function in the stress tensor calculation.

In some configurations, the method further includes defining the distortion strain as a logarithm of the stretch ratio of the simulated material according to the equation:

whereinEqual to pure shear at small strains,andis the true strain in the main direction,andis the stretch in the vertical direction along the simulated material and the distortion strain of the remaining surface is defined as the logarithm of the stretch ratio of the simulated material according to the following equation:

wherein。

In some configurations, the method further comprises calculating entropy elasticity using a cross-linked network in parallel with a generalized maxwell model, the maxwell element comprising a nonlinear spring storing energy as a volume ratio gibbs free energy, wherein the stress is derived according to the following equation:

。

in some configurations, the calculated stress is calculated in the principal orthogonal anisotropy direction according to the following equation:

。

drawings

The disclosure will now be described with reference to the accompanying drawings, in which:

FIG. 1 of the drawings illustrates an example simulation system, a version of which may include a control module, according to embodiments disclosed herein;

FIG. 2A shows an example of 3 diamonds on a cube deformed under tension according to embodiments disclosed herein;

FIG. 2B illustrates an example conventional cut on 3 planes referred to as 4, 5, 6 according to embodiments disclosed herein;

FIG. 2C shows an example of volume change from 3 orthogonal strains according to embodiments disclosed herein;

FIG. 3 illustrates an example shape of a Gibbs free energy curve versus shear strain and the resulting stress-strain relationship in accordance with embodiments disclosed herein;

FIG. 4 shows an example of a Morse potential energy function, such as a graphical Morse potential energy function versus strain, for affecting a volume free energy function, according to embodiments disclosed herein;

FIG. 5 illustrates an example generalized Maxwell model for volume or shear. According to embodiments disclosed herein, gibbs free energy changes may be tracked by monitoring strain in each of the springs shown;

FIG. 6 illustrates an exemplary general purpose computing device in accordance with embodiments disclosed herein;

FIG. 7 illustrates an example simulation system performing an example simulation of a material sample that is subjected to stretching in a vertical direction along a simulated material according to embodiments disclosed herein ₁And ₂；

fig. 8A and 8B show a first example graph including a distortion a strain and two curves (energy curve and shear stress curve) as an x-axis and a second example graph including a 1D strain and three curves (z, (respectively) as an x-axis, respectively, according to embodiments disclosed herein) Strain curves, morse energy curves, and morse stress curves);

FIG. 9 illustrates an example calculation of distortion in 6 axes and expansion in 3 axes according to embodiments disclosed herein;

10A and 10B show a first example graph including a distortion A strain and two curves (an energy curve and a shear stress curve) as an x-axis according to embodiments disclosed herein;

FIG. 11 illustrates a second example graph including 1D strain as the x-axis and three curves (z: (z) () Strain curves, morse energy curves, and morse stress curves);

FIG. 12 illustrates an example method 1200 for simulating strain-induced orthogonal anisotropy for a material, in accordance with embodiments disclosed herein; and

FIG. 13 illustrates an example method for simulating stress and strain for an orthotropic composite material in accordance with embodiments disclosed herein.

Detailed Description

While this disclosure is susceptible of embodiment in many different forms, there is shown in the drawings and will herein be described in detail specific embodiment(s), with the understanding that the present disclosure is to be considered as an exemplification and is not intended to be limited to the embodiment(s) illustrated.

It should be understood that throughout the drawings, similar or analogous elements and/or components referenced herein may be identified by similar reference numerals. Furthermore, it is to be understood that the figures are merely schematic representations of the invention and that some of the elements may have been distorted from actual scale for clarity of illustration.

Referring now to the drawings, and in particular to FIG. 1, there is disclosed a simulation system 10 that incorporates material science, thermodynamics, mechanics, and failure into a single process. Simulation system 10 includes simulation module 12, communication module 22, and programming module 24, each coupled to each other as shown. The simulation system 10 uses these new mechanics to make previously difficult problems manageable for the example disciplines:

nonlinear viscoelasticity-the properties of plastic and rubber materials vary with loading history, temperature and environment. The simulation system 10 simulates these viscoelastic materials. Alternative viscoelastic constitutive models may cover some narrow range of loading and environmental conditions. The simulation system 10 disclosed herein can handle any 3D loading and any temperature history.

Viscoelastic bond fracture-modern fracture mechanics cannot describe time and temperature dependent crack propagation in polymer bonding. The simulation system 10 disclosed herein can unify mechanics and fracturing into a single process, thereby achieving a solution.

Composite-unlike the prior art, the simulation system 10 disclosed herein can simulate glass and carbon fiber composites at the continuum (continuum) level, thereby speeding up computation time and also providing tracking of viscoelastic damage buildup.

Foam-closed cell polymer foams are particularly difficult to simulate. The simulation system 10 takes into account aerodynamic, localized, and microstructural effects that complicate the modeling of these materials.

Molecular dynamics amplification-quantum mechanics is used to simulate new material chemistry at the nanoscale. Simulation system 10 provides an unprecedented way to scale these nanometer results to the macroscopic scale.

The non-Newtonian fluid-simulation system 10 uses a unified theory that is also applicable to viscoelastic fluids, with important implications for tribology and polymer processing.

Plastic mechanics-the study of permanent deformation in metals is called plastic mechanics. The simulation system 10 clarifies the shear yield criteria and seamlessly integrates the failures caused by cavitation.

Biaxial testing-the new mechanics disclosed herein eliminate the need for complex, expensive, and often inaccurate biaxial experiments.

Impact physics-the new mechanics disclosed herein simplify shaped charge applications, such as those used in armor or petroleum industry casing perforation applications.

The simulation system 10 uses the new mathematical framework as part of an engineering simulation used in engineering product design. Examples of numerical simulations include finite element analysis, finite difference, and multi-volume simulations.

The simulation system 10 combines four isolated engineering disciplines into a single process: material science, thermodynamics, mechanics, and fracture/failure/plasticity mechanics. The simulation system 10 transfers the focus of mechanics from the stress-strain relationship to the free energy-strain relationship, revealing the unified theory of solid mechanics, hydromechanics and viscoelastic mechanics. The present disclosure begins with a relationship between thermodynamics and solid mechanics, and then integrates the concept of viscoelasticity, which in turn can be applied to fluid mechanics.

Mathematicians provide engineers with solid mechanics that relate stress to strain. The unified approach of the simulation system 10 instead relates free energy changes to strain. The stress is then calculated as the derivative of the free energy for each strain according to the following formula:

(1)

wherein the content of the first and second substances,is the 6 elements in the stress tensor,is 6 elements of the strain tensor, andis the change in gibbs free energy caused by mechanical deformation. Equation (1) is important because it is a bridge between thermodynamics and mechanics. Conventional mechanics typically directly relate stress to strain.

The simulation system 10 also includes a programming module 24. Programming modules 24 comprise a user interface of configurable simulation system 10. In many cases, programming module 24 includes a keypad with a display that is connected to control module 20 via a wired connection. Of course, programming module 24 may include a wireless device that communicates with control module 20 via a wireless communication protocol (i.e., bluetooth, RF, WIFI, etc.) using a different communication protocol associated with communication module 22. In other embodiments, programming modules 24 may include virtual programming modules in the form of software, for example, on a personal computer in communication with communication module 22. In still other embodiments, such virtual programming modules may be located in the cloud (or network-based), where they may be accessed through any number of different computing devices. Advantageously, with such a configuration, a user may be able to remotely communicate with the simulation system 10 with the ability to change functionality.

In at least one embodiment, simulator system 10 is coupled to manufacturing system 30. The manufacturing system 30 receives the simulation results generated by the simulation system 10 and manufactures one or more physical products based on the simulation results generated by the simulation system 10. For example, the manufacturing system 30 may manufacture any of the example products discussed herein, although other physical products are also contemplated.

The energy method has been practiced for centuries, but the present simulation system 10 introduces an important difference. The energy function is divided into 6 independent shear (distortion) and 3 correlation functions defining the volume (expansion). To visualize this, fig. 2A to 2C show cube elements representing deformations. In particular, fig. 2A shows 3 diamonds on a cube deformed under tension called shear 1, 2, 3, fig. 2B shows conventional shear on 3 planes called 4, 5, 6, and fig. 2C shows the volume change from 3 orthogonal strains.

Fig. 2A depicts the tension on the cube. In the undeformed state, the rhombuses start with 45 degree squares on each face. If the simulation material is initially isotropic (with the same properties in all 3 directions), the tension causes the rhombuses on both faces to twist into rhombuses, the squares on the faces perpendicular to the loading are still squares, undergoing expansion but zero distortion.

Fig. 2B represents one of three conventional shear strains. These shears are common to the embodiments disclosed herein and conventional mechanics. In contrast, fig. 2C represents something entirely unique compared to the conventional. This is one of the keys of the simulation system 10. As an example, for a uniaxial composite simulation material, hydrostatic pressure causes different strains in three different directions. The simulation system 10 assigns different material properties to the volume changes from each of the three different directions. Typically, the volume change is related to pressure by a single characteristic: bulk modulus.

In the most general case, the free energy function contains these 6 shear relationships and 3 volume relationships, as follows:

wherein the content of the first and second substances,is the change in the Gibbs free energy of the volume ratio, which is a function of the ith shear strain alone. The 6 energy to shear relationships are independent of each other. The change in Gibbs free energy due to the change in volume is a function of the logarithmic strain in 3 directions. The change in volume can affect 6 shear responses, particularly for simulated viscoelastic materials.

Combining equations (2) and (1) yields 6 stress-strain relationships:

for example, consider simulated orthotropic materials that have unique properties in 3 orthogonal directions. The continuous fiber composite may be orthotropic. For the limited case of small strain linear elastic orthonormal anisotropy, all energy functions are parabolic in nature:

wherein the content of the first and second substances,is the shear modulus of 6 pieces of rubber,is a shear strain of 6 or more,is a logarithmic strain, andare 3 characteristics related to the volume change. Combining (1) and (4) results in an orthotropic linear elastic stiffness tensor:

(5)

textbooks describing the mechanics of composite materials recognize that orthotropic materials require 9 independent properties, but they typically use 3 young's moduli, 3 poisson's ratios, and 3 shear forces. Equation (5) describes the upper left quadrant of the stiffness tensor in a form such that the 9 orthogonal anisotropy properties are 6 shears and 3 volumes.

Equation (5) is only valid for small strain linear elastic responses. Equations (2) and (3) are significantly more efficient because they are valid for all strains. The problem becomes what the shape of these energy functions isFirst consider the 6 shear energy relationships in the initially orthotropic material. The shear stress-strain relationship must satisfy 4 requirements.

1. Approximately linear at small strains, as defined by shear modulus;

2. antisymmetric, and therefore the same response for positive or negative shear strain;

3. triggering crack propagation or instability of plasticity (local maxima/minima); and

4. failure must eventually occur because at some non-zero strain the shear stress must be zero.

Fig. 3 shows the shape of the free energy curve versus shear strain and the resulting stress-strain relationship. The inverse gaussian distribution of the gibbs function satisfies these requirements, as shown in fig. 3. Note that the energy function may take any shape, as stimulated (or interpreted, i.e., motivate) by understanding material science, micro-mechanics, nano-mechanics, and/or Molecular Dynamics (MD) simulations. As disclosed herein, this is the way material science is linked to thermodynamics.

For bulk modulus, first consider the 1D morse potential function well known in material science. After conversion to strain, the function takes the form:

(6)

wherein the content of the first and second substances,Vis the potential energy of the human body,Eis the Young's modulus of the polymer,is the 1D engineering strain and c is a defining parameter.

The simulation material may be more complex than an orthotropic material (with different properties in 3 orthogonal directions). For example, the dummy material may be anisotropic or monoclinic. In addition, the microstructure may affect a response, such as a simulation result produced by the simulation system 10. Take a rope as an example. Axial forces can cause twisting as the rope attempts to unwind. In all of these cases, the gibbs free energy function may include additional terms to account for these types of simulation materials.

Fig. 4 shows a morse potential function, such as the illustrated morse potential function versus strain, for affecting the volumetric free energy function. The morse potential balances the repulsive and attractive atomic forces holding the two atoms together. When the atoms are pushed closer, the repulsive forces dominate and the dummy material becomes extremely hard. Under tension, the atoms eventually lose attraction and their connection eventually fails.

The simulation system 10 may use 1D morse to excite the 3D volumetric free energy function for the initially isotropic polymer:

(7)

equation (7) is the first step in establishing the volumetric energy function for a given simulated material and is intended as an example of this process. Like shear, the bulk free energy function may also be excited by material science, micro-mechanics, nano-mechanics, or molecular dynamics simulations.

The description so far describes solid mechanics, which means that there is no time dependence. The mechanical response of the dummy material may also have a viscous contribution. Mechanical simulations are useful tools for understanding the so-called viscoelastic response. Consider the generalized maxwell model mechanical simulation shown in fig. 5 used by simulation system 10. Fig. 5 shows a generalized maxwell model 500 for volume or shear. The gibbs free energy change can be tracked by monitoring the strain in each spring as shown.

The gibbs free energy function in the new mechanics of the simulation system 10 is applied to the spring in fig. 5. The time dependence then comes from the viscosity in the buffer. Each spring-damper pair is referred to as a maxwell element, and the maxwell elements can be combined in parallel to produce a discretized spectral response, representing stiffness as a function of time. For example, each spring-damper pair may fit a time-dependent master curve of the simulation material or describe the frequency response of the simulation material. Note that other functions besides the Prony series and maxwell elements can be used to obtain the time-dependent modulus.

The shortened time model is a viscoelastic constitutive model of the type used to simulate plastics, rubber and glass. In a reduced time, time dependencies are accelerated by loading history and context. For example, in time-temperature superposition, increasing the temperature accelerates the time-dependent response. The simulation system 10 accommodates all environmental conditions and mechanical loading history:

increasing the temperature for an accelerated time;

adding mechanical additivesVolume acceleration of the load for shear response;

increasing the free energy for each maxwell element accelerates the time for that element;

entropy in the rubber elastically slows down the time for all maxwell elements; and is

The solvent absorption accelerated time.

In tabular form:

one aspect of the simulation system 10 is the implementation of non-linear springs in maxwell elements. This innovation enables viscoelastic damage tracking and time and temperature dependent fracturing.

The simulation system 10 is versatile enough to encompass fluid mechanics, including viscoelastic fluids. To this end, the simulation system 10 simply removes the double spring on the left side of FIG. 5. The resulting unified theory explains all non-newtonian fluids: bingham fluids, shear thinning fluids, and shear thickening fluids. Just like viscoelastic solids, this theory involves the effect of pressure on fluid viscosity, which is important for tribological simulations and polymer processing.

Thus, according to the present disclosure, the simulation system 10 implements a numerical simulation based on 6 shear strains and 3 axial strains. The methods disclosed herein compare conventional 3-axial and 3-shear strain tensors and are effective with solid, fluid, or viscoelastic materials. In accordance with the present disclosure, it is possible to establish a small strain, orthotropic linear elastic stiffness tensor over 3 volumes and 6 shear moduli. Note that for small strains, linear isotropy, this is reduced to one volume and one shear. Numerical simulations may be performed based on six independent shear free energy-strain relationships and one volumetric free energy-strain relationship. The volume relationship is based on 3 orthogonal log strains. The methods disclosed herein are in contrast to direct stress-strain methods and are effective for large strains. Each shear free energy-strain relationship may be a function, some combination of functions, or a spline fit. It meets 4 criteria:

1. symmetric around zero strain (see fig. 3);

2. zero slope at large strain (i.e., failure at large strain);

3. approximate a parabola at small strain; and

4. the derivative has a peak that is unstable to drive failure.

For mathematical convenience, simulation system 10 may implement an inverted gaussian distribution for shear energy in accordance with the present disclosure. One aspect of the simulation system 10 disclosed herein is the form of the bulk modulus energy relationship:

wherein the content of the first and second substances,is thatA certain function of (a). This may be a function, a concatenation of functions covering a smaller range, or a cubic spline. For example, to generate the Morse potential excitation function in equation (7),

(9)

accelerated calculation time: a conventional non-linear solver must minimize 6 constitutive relations in parallel. Since the 6 strains are independent of each other, the 6 nonlinear equations are minimized one at a time and faster. The non-linear volume still needs 3-fold parallel minimization. Simulation system10 can implement 6 shear strains and 3 volume strains in Digital Image Correlation (DIC). Logarithmic strain has been reported in such DIC software, such as、And. Simulation system 10 may report according to FIG. 2、、And。

for nonlinear viscoelasticity, implementing the simulation system 10 disclosed herein results in a reduced time constitutive model for plastics and rubbers. A key contribution of the simulation system 10 is to use thermodynamic sub-states to accelerate time. To aid in understanding this, consider fig. 5. Instead of using the free energy of the simulation system to accelerate time, each individual Maxwell element has its own Gibbs free energy state. The thermal shift factor is measured directly and fitted with a spline curve. This approach eliminates the need for thermo-rheological simplicity. The volume change affects the viscoelastic shear modulus, although the 6 shear relationships are independent of each other. This is related to the pressure deceleration time of viscoelastic shear. Two springs are placed in parallel with the maxwell element, such as shown in fig. 5, one spring for entropy elasticity and the other representing a superelastic cross-linked network. Entropy elasticity slows down the time of shear relationship for all 6 viscoelastic spectra. This aspect of the simulation system 10 is critical to rubber simulation. Vertical displacement: the entropy elastic springs harden with increasing temperature and all other springs soften with temperature, as shown in fig. 5.

The parallelized superelastic spring shown in FIG. 5 may be a modified Gent/Arruda-Boyce:

1. different cross-linked network hardening parameters in uniaxial tension and compression;

2. for 3D loading, the hardening parameters are modified using the in-plane area;

3. multiplying by an exponential function to provide a peak in stress-strain (failure);

4. including foam-like positioning parameters; and

5. and carrying out Marins tracking through the shift network hardening parameters under shearing.

The change in entropy state plus the irreversible entropy produced by the buffer affects the heat produced by the mechanical loading. The resulting heat changes temperature, which can be fed back into the mechanical loading and change response.

The viscoelastic volume response is included in the model. The alternative model assumes incompressibility.

For the fracture and failure criteria, the 6 shear and 3 volume stress-strain curves have peaks. This peak represents instability, which can be used to mathematically trigger crack propagation. The nonlinear springs in the generalized maxwell model of fig. 5 can account for time and temperature dependent composite bond fractures. The addition of a damage state variable to each maxwell element (MWE) can effect viscoelastic damage accumulation in fatigue fracture. As each MWE reaches the peak of the stress-strain response of the spring, its ability to reach that peak will deteriorate. Unlike the present situation, the failure may also be distorted or dilated. Degradation may change the peak strain, but still return to the initial zero strain at zero stress. This is the way in which volume damage may respond, although volume compression is not affected by the accumulation of damage. Degradation may change the initial strain at zero stress, resulting in permanent deformation. This is a reasonable case for shearing. Note that shear requires twice the state variables within the damage, one for positive strain and one for negative strain, thereby linking free energy to strain.

The strain energy density is the fundamental stone of traditional fracture mechanics. The strain energy release rate (G) is a fracture prediction material property. The well-known J-integral determines the energy at the crack tip by measuring the area integral of the energy of the structure around the crack. But the traditional fracture mechanics derives from traditional mechanics, which relates 6 stresses to 6 strains. The simulation system 10 combines solid mechanics and fracture mechanics into unified mechanics.

The simulation system 10 may replace a cohesion model (a coherent zone model), which is another fracture simulation method in finite element analysis for predicting composite fractures, particularly for cohesive fractures. This method defines the mode I (opening) and mode II (shear) drag separation (TS) laws for crack propagation. In the current state of the art, TS law cannot capture rate, time or temperature dependencies. The simulation system 10 integrates the TS-type law into a nonlinear spring, using distortion and expansion instead of mode I and mode II. Thus, the simulation system 10 naturally accommodates the effects of time/temperature on polymer adhesion.

Fatigue fracture models also exist, particularly the model implemented by endrrica. These models assume that cracks open only in mode I, cannot correctly track the heat accumulation caused by mechanical cycling, and typically ignore the effects of temperature and rate on viscoelastic material properties.

For composites, simulation system 10 provides viscoelastic expansion damage accumulation in polymer matrix composites, viscoelastic distortion damage accumulation in polymer matrix composites, and applies 9 properties of 6 shears and 3 volumes to orthotropic composites on a continuum level.

It is known that a linear elastic orthotropic material requires 9 independent material properties. Traditional composite textbooks use 3 young's moduli, 3 poisson's ratios, and 3 shear moduli. These properties lead to the following conclusions: in orthotropic materials, expansion cannot be separated from distortion, a conclusion that is fundamentally in conflict with strain-induced orthotropic.

A relatively recent development in composite simulation is the micromechanical simulation used to define macroscopic material properties. Software code like Digimat or MultiMechanics attempts to simulate small scale interactions between the substrate and the reinforcement material. These methods are computationally very time consuming and are poor at tracking viscoelasticity and damage. The simulation system 10 provides a continuum-level orthotropic solution of viscoelastic damage accumulation. Furthermore, failure may occur in distortion or swelling.

For plastic mechanics, the well-known von mises failure criterion stems from maximum distortion. The simulation system 10 instead tracks the 6 shearing relationships independently.

A bistable energy state function for volume can be used to trigger neck instability in polymers. In expansion tension, the proposed energy curve will have a second local minimum.

The spectroscopic method commonly used for viscoelasticity (fig. 5) is also applicable to plastic mechanics. Multiple parallel mechanisms can lead to failure. These parallel mechanisms do not necessarily need to be viscoelastic. The energy-strain response of each mechanism will be inspired by material science. For example, carbide formation under certain expansion strains may alter the contribution of the carbide to the overall load response. As another example, a parallel mechanism may describe cracks growing in partially stabilized zirconia. The prior art for plasticity does not consider 6 independent shear relationships, each of which can potentially be affected by swelling.

For hydrodynamics, such as shown in fig. 5, without the two springs on the left, a shear-thinning fluid can be simulated under constant strain rate loading with reduced plateau stress because the maxwell elements increase in energy sub-states over gibbs, accelerating time and reducing the strain rate for reduced time. For bingham fluids, a weakly cross-linked network like marlins in rubber must be overcome before flow can begin. Unlike rubber, there is no cross-linking of chemical bonds to prevent flow.

For tribology, the simulation system 10 incorporates hydrostatic pressure into viscoelastic fluid behavior. For shear thickening fluids, the softening by gibbs free energy has less effect than the hardening by the increase in strain rate in real time.

Viscosity is the primary material property in fluids compared to the modulus (i.e., stiffness) used in solids. Viscosity varies with strain rate, but at a given strain rate it does not vary with time. In other words, viscosity is not a functional strain history, just like viscoelastic modulus. This complicates viscoelastic fluid constitutive modeling.

Classical hydrodynamics takes into account shear viscosity and extensional viscosity. The simulation system 10 considers 6 cuts. Furthermore, hydrodynamics are classically considered incompressible, ignoring the viscoelastic volume response.

Finally, non-Newtonian fluids are classified into different categories according to their own constitutive laws. These include shear-thinning fluids, shear-thickening fluids, and bingham fluids. The simulation system 10 unifies all of these into a single theory.

For molecular dynamics amplification, the simulation system 10 uses MD simulation to guide the shape of the energy function. Fig. 3 and 4 show points from molecular dynamics simulations. This work was done with schroederinger, but the simulation system 10 integrated these results into a continuum level model. MD is limited to extremely short time scales. The viscoelastic simulation disclosed herein can compensate for this short time by simulating a short time stress relaxation test at high temperatures, as temperature accelerates time. In this way, the simulation system 10 predicts the time and temperature response of the simulated polymer.

MD simulation implements virtual chemistry, enabling material companies to iterate through multiple iterations quickly, and mitigating safety concerns of producing unknown compounds. Unfortunately, MD mechanical simulations require a significant amount of computational time, e.g., microsecond events on a 40 nm polymer cube can take several days to simulate. Scaling to continuum level is considered the industry's "holy grail". The simulation system 10 provides this amplification.

For foams, local buckling on the microstructural length scale complicates foam simulation. The nonlinear springs in the generalized maxwell model (as shown in fig. 5) can be customized to include an initial local peak and a plateau caused by microstructural buckling instability. The mechanical response of closed cell foams can be dominated by aerodynamic contributions. The volumetric contribution to stress may include an isotropic pneumatic spring whose response is based on the ideal gas law. The mechanical response of closed cell foams can be dominated by aerodynamic contributions. The volumetric contribution to stress may include an isotropic pneumatic spring whose response is based on the ideal gas law. When modeling foam, viscoelasticity and closed cell response are generally ignored or at least avoided. Simulation system 10 makes these problems manageable.

For multi-body simulations, simulation system 10 may be combined with finite element analysis to build a library of components for multi-body simulations to enable time and temperature dependence in rubber bushings and tires. The limitations of rubber bushings and tires are a well-known problem of multi-body dynamics simulation software. Some libraries exist, but they generally do not take into account frequency or temperature dependence. The simulation system 10 may be used in conjunction with Finite Element Analysis (FEA) to build a library of suitable nonlinear visco-elastic properties.

The simulation system 10 simulates material characteristics and failure criteria. Emulation is a powerful tool for product development because products and their subsystems can be virtually tested. Virtual prototypes enable many more design iterations in much less time and at significantly lower cost. The simulation speeds up time to market, improves quality, and reduces development costs.

Engineering simulations are built on three basic columns: 1) defining geometry, 2) applying boundary conditions, and 3) defining material constitutive relations. The simulation system 10 is centered on the column 3. The simulation system 10 is suitable for finite element analysis (typically applied to solids), finite difference methods (typically applied to fluids), and other solution methods.

Consider a specific example of Abaqus finite element software, where simulation system 10 is a user-defined material model, referred to as UMAT. UMAT calculates the stress from the strain from Abaqus. It also computes the jacobian, which is required by the non-linear solver of Abaqus. The simulation system 10 internally implements new mechanics, receiving strain and returning stress in the traditional 6-element tensor format. The simulation system 10 is a material definition that includes a failure criterion.

Solid mechanics is established by the 6-element stress tensor and the 6-element strain tensor. In finite element analysis, the solver solves all six elements of each finite element in the model simultaneously. The simulation system 10 implements 9 mathematical relationships. Not coincidentally, classical mechanics requires that orthotropic materials have 9 independent material properties, with orthotropic materials having unique properties in all 3 orthogonal directions. The simulation system 10 relates each shear strain to only one shear stress by a relationship, which speeds up the calculation.

The simulation system 10 is not the first system to relate energy to mechanics. Notably, superelastic rubber models such as Gent or Arruda-Boyce use so-called strain energy density functions. The strain energy density should be called the Gibbs free energy change of volume ratio. Nevertheless, energy has been previously defined from the perspective of strain energy invariants. The simulation system 10 divides the free energy into 6 independent shears and one volume. Thus, there are 9 independent relationships linking stress to strain, similar to the 9 material properties in linear elastic orthotropic anisotropy.

For nonlinear viscoelasticity, there are many constitutive models of polymers. Mechanics tend to be built on strain invariants rather than the 9 strains of the simulation system 10. Many of these viscoelastic models are not compressible and cannot capture strain-induced anisotropy. All models cover some narrow range of temperature and loading history conditions, but none describe the complete response particularly well. Some poor, competitive constitutive models include Bergstrom-Boyce, Parallel theoretical frame, Potential Energy Clock, and free volume.

Referring to FIG. 6, an apparatus, such as an exemplary general purpose computing device, is disclosed for performing the simulations described herein. The general purpose computing device is shown in the form of an exemplary general purpose computing device 100. The general purpose computing device 100 may be of the type used for the control module 20 (FIG. 2). As such, the description will be with the understanding that variations thereof are possible. The exemplary general purpose computing device 100 can include, but is not limited to, one or more Central Processing Units (CPUs) 120, a system memory 130, and a system bus 121 that couples various system components including the system memory to the CPU 120. The system bus 121 may be any of several types of bus structures including a memory bus or memory controller, a peripheral bus, and a local bus using any of a variety of bus architectures. Depending on the particular physical implementation, one or more of the CPU120, system memory 130, and other components of the general purpose computing device 100 may be physically co-located, such as on a single chip. In this case, some or all of the system bus 121 may simply be a communication path within a single chip structure, and its illustration in FIG. 6 may be merely a convenient label for illustration purposes.

General purpose computing device 100 also typically includes computer readable media, which can include any available media that can be accessed by computing device 100. By way of example, and not limitation, computer readable media may comprise computer storage media and communication media. Computer storage media includes media implemented in any method or technology for storage of information such as computer readable instructions, data structures, program modules or other data. Computer storage media includes, but is not limited to, RAM, ROM, EEPROM, flash memory or other memory technology, CD-ROM, Digital Versatile Disks (DVD) or other optical disk storage, magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices, cloud data storage resources, video cards, or any other medium which can be used to store the desired information and which can accessed by the general purpose computing device 100. Communication media typically embodies computer readable instructions, data structures, program modules or other data in a modulated data signal such as a carrier wave or other transport mechanism and includes any information delivery media. By way of example, and not limitation, communication media includes wired media such as a wired network or direct-wired connection, and wireless media such as acoustic, RF, infrared and other wireless media. Combinations of any of the above should also be included within the scope of computer readable media.

When using communication media, the general purpose computing device 100 may operate in a networked environment via logical connections to one or more remote computers. The logical connection depicted in FIG. 6 is a general network connection 171 to a network 190, which network 190 may be a Local Area Network (LAN), a Wide Area Network (WAN) such as the Internet, or other network. The computing device 100 is connected to the general network connection 171 through a network interface or adapter 170, which network interface or adapter 170 is in turn connected to the system bus 121. In a networked environment, program modules depicted relative to the general computing device 100, or portions or peripherals thereof, may be stored in the memory of one or more other computing devices that are communicatively coupled to the general computing device 100 through the general network connection 171. It will be appreciated that the network connections shown are exemplary and other means of establishing a communications link between the computing devices may be used.

The general purpose computing device 100 may also include other removable/non-removable, volatile/nonvolatile computer storage media. By way of example only, FIG. 6 illustrates a hard disk drive 141 that reads from or writes to non-removable, nonvolatile media. Other removable/non-removable, volatile/nonvolatile computer storage media that can be used with the exemplary computing device include, but are not limited to, magnetic tape cassettes, flash memory cards, digital versatile disks, digital video tape, solid state RAM, solid state ROM, and the like. The hard disk drive 141 is typically connected to the system bus 121 through a non-removable memory interface such as interface 140.

The drives and their associated computer storage media discussed above and illustrated in FIG. 6, provide storage of computer readable instructions, data structures, program modules and other data for the general purpose computing device 100. In FIG. 6, for example, hard disk drive 141 is illustrated as storing operating system 144, other program modules 145, and program data 146. Note that these components can either be the same as or different from operating system 134, other program modules 135, and program data 136. Operating system 144, other program modules 145 and program data 146 are given different numbers here to illustrate that, at a minimum, they are different copies.

The embodiments discussed above include hyperbolic secant functions, where angles are used to define distortion strains. According to at least one other embodiment, at least one embodiment discussed below defines the distortion strain as the natural logarithm of the stretch ratio. This new distortion strain definition eliminates hyperbolic secants, clarifying the strain definition. According to embodiments disclosed herein, at least nine (9) independent mathematical relationships are utilized to define their energy functions. Typical functions separate expansion and distortion, but they use the distorted J2 strain invariant and the first principal strain invariant (I1) to define the energy. Thus, a typical function utilizes only two mathematical parameters, where the embodiment(s) disclosed herein simulate orthogonal anisotropy based on at least nine independent mathematical relationships. In the case of strain-induced orthotropic, the energy of the simulated isotropic material is defined in the direction of the main strain. This means that three (3) of the mathematical relations are three (3) main directions. Three distortions are then defined in these directions. The 3 expansion z-functions are also defined by 3 principal strains. Otherwise, the energy varies with the selection of the reference direction.

In accordance with at least one other embodiment, which builds upon the embodiments disclosed above, a simulation system 10 is disclosed that utilizes new mechanics including a new strain definition of an energy function. This new strain definition first defines new strain(s), then defines an energy function from these strains, and then calculates the stress as the derivative of the energy. Advantages of this new mechanics include that it separates expansion and distortion, even for orthogonal anisotropy, e.g., strain induced orthogonal anisotropy, it ties thermodynamics and mechanics together and provides a basis for solving complex problems such as nonlinear viscoelasticity, composite fracture, viscoelastic damage in composites, rubber mechanics, plastic mechanics, tribology, etc.

Referring to FIG. 7, the simulation system 10 simulates a material sample 700, the material sample 700 being subjected to stretching in a vertical direction along the simulated material ₁And ₂. For the first side, simulation system 10 calculates pure shear as the logarithm of the stretch ratio according to the following equation:

wherein the content of the first and second substances,equal to pure shear at small strain, and whereinAndis the true strain in the main direction.

Also, the simulation system 10 more generally calculates pure shear at small strain for the remaining two planes using the following equation:

wherein。

The simulation system 10 defines the expansion energy from the main real strain as visualized in fig. 2C. The simulation system 10 calculates the isotropic small strain using the following formula:

the simulation system 10 further calculates the large strain using the following equation:

where z is the expansion function and ε is in the main strain direction. For simulations of stress and strain for orthotropic composites discussed in more detail below, ε is the strain in the main direction of the orthotropic, κ is the bulk modulus, and the z-function combines to contribute to the expansion of the free energy.

Thus, the expansion can be defined as the sum of three z-functions, where each z-function depends on only one orthogonal strain. For example, in the case where an iso-triaxial load is applied to an anisotropic cube, different stresses are required in 3 orthogonal directions. The derivative of the volumetric energy provides these unique stresses.

The simulation system 10 also defines the strain energy density in the primary direction for the initially isotropic material according to the following equation:

where b is a vertical shift factor that is a function of the expansion, linking expansion to distortion.

The system 10 then calculates the stress from the energy according to the following equation:

referring to fig. 8A and 8B, example shapes of the energy function are shown to include at least nine (9) relationships for strain-induced orthogonal anisotropy calculated by the simulation system 10. In particular, fig. 8A shows a first graph 810 including a distortion a strain and two curves (an energy curve 812 and a shear stress curve 814) as the x-axis. FIG. 8B shows a second graph 820 including a 1D strain as the x-axis and three curves (z:)) Strain curve 816, morse energy curve 818, and morse stress curve 822).

The simulation system 10 calculates the stress in the principal direction based on the calculated distortion stress according to the following equation:

for example, for an isotropic sample under uniaxial loading, if the volume and shear energy functions are known, the simulation system 10 can predict linear elasticity, plasticity, and fracture using the following equations:

therefore, as shown by these equations, the expansion must balance the distortion in the lateral direction.

According to at least one other embodiment, which builds upon the embodiments disclosed above, the simulation system 10 utilizes new mechanics including new strain definitions, energy, and composite materials. For the simulated material that is already orthotropic, the simulation system 10 continues to use 6 shear strains and 3 volume strains. Similar to that discussed above, this new strain definition first defines new strain(s), then defines an energy function from those strain(s), and then calculates the stress as the derivative of the energy. Advantages of this new mechanics include that it completely separates expansion and distortion, even for orthogonal anisotropy, it links thermodynamics and mechanics together, and provides a basis for solving complex problems such as MD magnification, viscoelastic damage accumulation, cavitation failures, thermoplastic self-healing, polymer handling and thermoplastic flow, and environmental effects (e.g., temperature, solvent).

Referring to FIG. 9, simulation system 10 calculates the distortion in 6 axes and the expansion in 3 axes, as shown. For the simulated orthotropic material, simulation system 10 calculates the small strain based on nine (9) independent material properties and calculates the large strain based on nine (9) unique stress-strain relationships that are valid for the large strain. Simulation system 10 calculates the energy to convert the 9 unique relationships into a six (6) -membered stress tensor according to the following equation:

as shown in FIG. 7, the simulated orthotropic material underwent stretching ₁And ₂. As discussed above, the simulation system 10 calculates the pure shear for the simulated orthotropic material as the logarithm of the stretch ratio. As discussed above, the simulation system 10 also defines the expansion energy for the simulated orthotropic material from the dominant true strain. The simulation system 10 also defines the strain energy density in the principal direction for simulating orthotropic materials according to the following equation:

where b is the vertical shift factor as a function of the expansion, linking expansion to distortion.

The simulation system 10 also calculates simulated orthotropic material stresses from the energies according to the following equation (as discussed above):

referring to fig. 10A and 10B, example shapes of energy functions for simulated orthotropic materials are shown to include at least nine (9) relationships of strain-induced orthotropic as calculated by simulation system 10: six (6) distortion strains and three (3) expansion strains. In particular, fig. 10A shows a first graph 810 comprising a distortion a strain as the x-axis and two curves: an energy curve 1012 and a shear stress curve 1014. FIG. 10B shows a second graph 1020 that includes the 1D strain as the x-axis and three curves: z: () Strain curve 1016, morse energy curve 1018, and morse stress curve 1022.

As discussed above, the simulation system 10 calculates the stress in the principal direction for the simulated orthotropic material based on the calculated distortion stress. For isotropic samples that simulate orthotropic materials under uniaxial loading, the simulation system 10 can also predict linear elasticity, plasticity, and fracture using the equations disclosed above if the volume and shear energy functions are known.

Referring to fig. 11, a maxwell model 1100 for simulating orthotropic and nonorthogonal anisotropic materials is shown with nonlinear viscoelasticity, as used by the simulation system 10. Maxwell model 1100 calculates entropy elasticity using a cross-linked network. Maxwell model 1100 includes a nonlinear spring that stores energy as a change in Gibbs free energy, where stress is the derivative according to the following equation:

maxwell model 1100 includes nonlinear viscoelastic (NLVE) response Eying Polanyi shortening time according to the following equation:

fig. 12 illustrates a method 1200 for simulating strain-induced orthogonal anisotropy for a material. Method 1200 is stored on a non-transitory storage medium (e.g., ROM131 and/or hard drive 141) and executed by a processor, such as CPU 120.

The method 1200 includes a process 1210 of calculating three (3) principal strain directions for the simulation material. Process 1210 advances to process 1220.

Process 1220 includes calculating three (3) distortion strains for the simulated material. Process 1220 proceeds to process 1230.

Process 1230 includes calculating three (3) expansion strains for the simulation material. Process 1230 proceeds to process 1240.

Process 1240 includes calculating a free energy for the simulated material, the calculated free energy calculated from the calculated three principal directions, the three distortion strains, and the three expansion strains of the simulated material. Process 1240 advances to process 1250.

Process 1250 includes calculating, via the calculated free energy, a stress for the simulation material based on the calculated free energy for the simulation material.

FIG. 13 illustrates another method 1300 for simulating stress and strain for an orthotropic composite material. Method 1300 is stored on a non-transitory storage medium (e.g., ROM131 and/or hard disk drive 141) and executed by a processor, such as CPU 120.

Method 1300 includes a process 1310 of calculating six (6) distortion strains for a simulated orthotropic composite material. Process 1310 proceeds to process 1320.

Process 1320 includes calculating three (3) expansion strains for the simulated orthotropic composite material. Process 1320 proceeds to process 1330.

Process 1330 includes calculating a free energy for a simulated orthotropic composite, the calculated expansion energy calculated from six distortion strains and three expansion strains. Process 1330 advances to process 1340.

Process 1340 includes calculating, via the calculated free energy, a stress for the simulated orthotropic composite material based on the calculated expansion energy for the orthotropic material.

The foregoing description merely illustrates and explains the disclosure, and the disclosure is not limited thereto except insofar as the appended claims are so limited, as those skilled in the art having the disclosure before them will be able to make modifications without departing from the scope of the disclosure.