Luận văn thạc sĩ Kỹ thuật cơ khí: Assessment of energy absorption of foam material for applications in fall protection by finite element analysis

OVERALL ABOUT RESEARCH THEME

The urgency of theme

PU Foam (known as Polyurethane Foam) and sponge rubber are compressible, resilient, lightweight materials that help insulate, absorb shock and impact, damp noise and vibration, and seal gaps and joints Foams can be closed-cell, more suited to sealing applications, or open-celled, which are typically better for damping applications Both come in a vast range of base chemistries, densities, thicknesses, and additional properties to help meet specific product requirements

Figure 1.1 Example for stress-strain curve of PU Foam material

Foam functions in various forms such as insulation foam rolls, custom foam sheets, protective foam packaging, molded foam products and specially made foam products for individual industry or application

Foam materials are typically categorized into two types, closed cell foam and open cell foam Both types of foam materials provide superior cushioning and protective performance The most important aspect of foam material advantage is cushion and protection compared with other rigid plastic and wood materials Some of the most common types are EVA foam (also known as Ethylene-vinyl acetate copolymer foam), polyethylene foam, polyurethane foam and foam rubber materials

Foam is the outcome of presenting gas bubbles into a polymer It’s a permeating, lightweight material and it comes in a wide range of types Simply speaking, foam material is version of expanded plastic and rubber They are usually resilient with high elasticity which makes foam ideal to deliver good cushioning and supporting for other products

Figure 1.2 Example for Foam Material

With a foam material, less is sometimes more Employing foam knowledgeably and professionally denotes being economical with it, to boost its efficacy Through shipment, a package will be susceptible to various types of hazards A foam will be efficient in lowering the impacts of various types of hazards like compression, vibration and impact

When combined with corrugated packaging, a foam material is a wonderful alternative for safeguarding items while they are being transported Foam material enables the objects to arrive at their location safely and make a better first impression for consumers once they unbox the product

Figure 1.3 One of Applications for Foam Material

In actual usage, the cushioning and protection advantage or benefit reflects in many applications and products For example, foam materials are custom fabricated into various sizes of foam packaging insert or foam pad, foam gaskets They are widely used for protecting or transporting fragile or expensive products Also you can find a lot of cushioning foam play mats for baby and children play at home or outdoor area, and comfortable foam neck pillows, foam seat cushion pads and so much

Foam is widely used in the construction industry to help insulate people’s homes There are many reasons why foam is used in this way First it can help block outside noise, keeping the house quieter It is also used to manage the temperature of the house Foam can help keep the house cool in the summer and warm during the winter months This can save you money, by making it more affordable to keep your home at a pleasant temperature However, these thermal insulation properties aren’t just limited to the construction site

Figure 1.4 One of Applications for Foam Material

In addition, foam is also commonly used to produce camping outdoor products Foam is lightweight, making it easy to carry when hiking Camping sleep mats are made of foam material and just take advantages of excellent heat insulation feature of foam Foam mats can effectively keep users ware during outdoor environment When used in matting or sleeping bags, foam can help keep campers warm For example, on a cold night, the person’s body heat will be captured by their foam sleeping mats This will keep them warm throughout the night, preventing them from developing conditions like hypothermia Some of sleeping mats are also made with aluminum foil back to enhance insulation ability.

Research status of PU Foam in the country and abroad

1.2.1 Research status of PU Foam in the country [1]

In recent years, polyurethane foam (PUF) has been known for its many preeminent properties: low density, high thermal and sound insulation properties, high strength, good physical-mechanical properties, highly recyclable and reusable, etc [2,3] These unique properties of PUF have led to a wide range of applications for this material PUF is applied in many different fields, such as making furniture in the aerospace industry, making home furniture, laptop screen protectors, outer cases for mobile electronic devices, wheelchairs, power tools, sporting goods, drive belts, shoes, inflatable rafts, and various extruded films and sheets, cushioning material, carpeting, furniture, bedding, automotive dashboards, automotive interior details, packaging in medicine, food technology, biomedicine and nanocomposites medical devices, heat and sound insulators in the construction, electronics, and transportation industries, artificial hearts, pacemaker and hemodialysis tubes, coatings, adhesives, sealants, binders, as an adsorbent in wastewater treatment, etc [4,5] The high applicability of this material has led to an increasing demand for them According to Fortune Business Insights' statistical report on the global polyurethane market size, the global polyurethane foam market reached 60.5 billion USD in 2017; 114.8 billion USD in

2019, and is expected to reach 157.63 billion USD in 2026 [6] The rapid increase in demand for PUF requires the supply of many raw materials for synthesis As is known, the two main components for PUF synthesis are isocyanate and polyol, which are obtained from petroleum Due to the increasing energy demand, oil petroleum as a fuel source is also increasing However, this resource is a non-renewable source and is increasingly depleted Besides, the exploitation of this energy source leads to significant environmental pollution [8,9] Therefore, the use of oil-based raw materials tends to be limited and undesirable In addition, the cost of polyol and isocyanate components derived from petroleum is relatively high To solve the above problems, the current trend of the PUF industry is to use natural fillers to reduce costs and limit the use of petroleum- based raw materials [2,7]

With this trend, chitin has been used as a naturally derived filler It is the second most abundant natural polymer in nature after cellulose It has been shown that the structure of chitin contains two hydroxyl functional groups, which can chemically interact with the isocyanate group to form urethane bonds, similar to the reaction between polyols and isocyanates [11] These theoretical bases have developed polymer composite material (PCM) based on PUF and chitin (PUFCs) as an adsorbent [5,10] The obtained PUFC material possesses the advantages of both raw materials: PUF and chitin, such as high adsorption and buoyancy [10,12], high recyclability, reuse, and oil recovery [13] Furthermore, it has been demonstrated that chitin filling into the PUF matrix significantly reduces the cost of the original PUF [5] Calculation of economic benefits shows that the cost of composites PUFC is reduced by USD 870/ton compared to PUF without filler [5] However, this polymer composite material has only been studied and investigated for its properties in wastewater treatment contaminated with oil and heavy metal ions [5,12] Still, it has yet to be evaluated for application in other fields, such as building materials, furniture, and the automotive industry, or applications in other sectors With the increasing pollution of solid waste and the depletion of petroleum resources, a material that can be applied in many fields will be superior Therefore, it is necessary to evaluate other properties of PUFC so that there are suitable application proposals not only for wastewater treatment with this potential material

However, using a multifunctional material in large quantities inevitably releases much solid waste into the surrounding environment Therefore, it is necessary to propose an alternative for dealing with consumed PCM Because the PCM material is based on the PUF matrix, its outstanding property is its low density This makes treating solid waste by traditional methods such as landfilling, incineration, or biological composting difficult Low-density materials will require an extensive landfill area when treated by these methods Not only is this not feasible, but it also has a significant negative impact on the environment, such as gas generation when landfilled or burned [14] Therefore, a reasonable and environmentally friendly recycling method for spent materials is essential

In this work, to suggest other applications for chitin-grafted-PUF materials, in addition to investigating the technological parameters in the foaming, the physical- mechanical properties such as compressive stress, compression set, and resilience are also evaluated Besides, the cross-link density of composition also is studied Furthermore, the morphology of the optimal material is also examined by SEM analysis to explain the changes in the physical-mechanical properties of obtained materials At the same time, in this research, waste- containing PCM is also developed with the purpose of recycling spent materials adhering to the principles of a circular

1.2.2 Research status of PU Foam in the abroad [15]

The history of polyurethane foams began in earlier 19th century with a German scientist named Otto Bayer Pieces of literature report that he was the first to synthesize foams using polyurethane He reported that PU foams are special polymer foams with many advantages over existing ones [16,17] Generally, polyurethane foams discuss a class of polymers primarily composed of organic chemical units joined by urethane links Though there are large number of existing polymer foams such as polystyrene, polyethene, and PU foams have attracted many researchers and scientists due to some unique features such as low density, flexibility and lightweight A recent survey reported that PU foams occupied

25 million metric tonnes of global production in 2020 [18]

Conventional PU foams are produced by reacting two raw materials, polyols and isocyanates, by polymerization process under catalysts and blowing agents, as shown in Figure 1.5 The most commonly used isocyanates are toluene diisocyanate (TDI) and diphenyl diisocyanate (MDI), as they are less expensive when compared to other isocyanates [19,20] Similarly, polyols which were used for the synthesis of conventional

PU foams, are petroleum-based chemicals But recently due to the rapid depletion of fossil fuels and toxicity, biobased polyols such as water-based castor oil-based are developed to overcome this phenomenon [21] In addition to the above raw materials, such as catalysts and surfactants, are added to develop special varieties of foams which can regulate variation in cell size, structure and density of the foam [22]

Figure 1.5 Process and formation of polyurethane foam

One of the notable characteristics of polyurethane foam is that they are viscoelastic material, which means it tends to regain its original shape under extreme force, making them superior over compression applications [23] Polyurethane foams were considered to be one of the first core materials used in sandwich structures Based on physical properties such as density, cell structure, cell size and orientation, foams can be broadly classified into two types: open and closed cells [24], as represented in Figure 1.5 Polymer foams are classified into rigid, flexible and semi-rigid foams, as shown in Table 1.1 [25]

Table 1.1 Classifications of foams in terms of Elastic Modulus

Analyze the advantages and disadvantages of the approach

• The approach is easy for user because just choosing Low Density Foam model and providing the stress-strain curve in Abaqus software, it can be used for PU Foam

• Microstructure of PU Foam is very complex in reality but we only need to simulate as a solid and assign suitable material property in simulation

• Foam behavior result between Simulation and Testing is quite matching (deviation smaller 10%) based on chapter 4.1 and 4.3

• The stress-strain curve from this approach is only exactly and applied for compressive Foam With tensile Foam, need to consider other approach

• PU Foam is only used for simple shapes

• Must have Specific Energy Absorption – Linear compression graph from material datasheet to calculate and convert into the stress-strain curve.

The purpose of theme

Concentrating research to use Foam material model in Simulation so that the behavior of Foam in Simulation has to match with actual test (the deviation is allowed below 10%) based on G-Force graph between Simulation and test data From that, we have a certain reliability to evaluate components in a product whether they have safe or not?

The research scope

Due to having a lot of Foam types and different characteristics of them so in this theme, I only focus on the PU Foam and useful its applications in Simulation as well as reality.

The research content

• Only focus on PU Foam

• Use suitable material model for PU Foam (Low Density Foam)

• Compare simulation result for PU Foam model with article based on compression test

• From material datasheet of customer provided, how to convert and calculate to have stress-strain curve for PU Foam material

• Apply Finite Element Method (Dynamic Explicit) to evaluate behavior of a product or identify Acceleration values, stress/strain on components

• Perform some reality problems with Foam material to see how good is PU Foam Material

• For reality product, compare Simulation result and test data (the deviation is allowed below 10%) base on Acceleration values, from that we have a certain reliability to evaluate components in a product whether they have safe or not?

The research method

• The Hyperelastic material model theory, Finite Element Method and the Dynamic Explicit Method are applied and researched in this thesis scope.

THEORETICAL BASIS

Theoretical basis of Hyper-elastic

2.1.1 Theoretical basis of Neo-Hookean model [26] [27]

The strain energy potential can be offered (Bol and Reese, 2003):

The Neo-Hookean model is obtained as follows:

+ W: Sometimes is written as U is the strain energy density or stored energy function defined per unit volume

+ 𝜇 1 : The shear behavior of the material

+ U: The strain energy potential (or strain energy density), that is the strain per unit of reference volume

+ I̅ 1 : The first invariants of the deviatoric strain

+ J el : The elastic volume ratio

+ 𝐷 1 : Introduces compressibility and is set equal to zero for fully incompressible materials

Under uniaxial extension, 𝜆 1 = λ and 𝜆 2 = 𝜆 3 = 1/√λ Therefore,

𝜎 11 − 𝜎 33 = 2𝐶 1 (λ 2 − 1 λ); 𝜎 22 − 𝜎 33 = 0 (2.3) Assuming no traction on the sides, 𝜎 22 =𝜎 33 =0, so we can write:

2.1.2 Theoretical basis of Mooney-Rivlin model [26] [27]

Strain energy potential is proposed:

Where 𝐶 𝑖𝑗 are material parameter and 𝐶 00 = 0 (Markmann and Verron, 2006) The first order for incompressible materials is presented as follows:

U = 𝐶 10 (I̅ − 3) + 𝐶 1 01 (I̅ − 3) 2 (2.6) The Mooney-Rivlin model is obtained as follows (Sasso et al., 2008; Toth et al., 2005):

+ U: The strain energy potential (or strain energy density), that is the strain per unit of reference volume

+ 𝜇 1 , 𝜇 2 : The shear behavior of the material

Then the expressions for stress become:

+ I̅ 1 , I̅ 2 : The first invariants of the deviatoric strain

+ W: Sometimes is written as U is the strain energy density or stored energy function defined per unit volume

2.1.3 Theoretical basis of Low Density Foam model [28]

The stress–strain relationship of a Polyurethane (PU) elastomer in uniaxial compression is nonlinear: the curve gradients increase as the strain magnitude increases

Apart from the foam model, all other models consider the material to be incompressible or minimally compressible As a result, the foam model was applied to model the PU because of its high compressibility Although the microstructures of foams are irregular on the cellular scale, they can be considered to be a homogeneous continuum within a certain area on a large scale

Figure 2.1 Stress versus strain curves of low-density foam

The foam model is a modification of the Ogden model for highly compressible low- density foams If thermal expansion can be ignored, the elastic behavior of the foam model is based on the strain energy function:

Where N is the potential order of the strain energy and is an integer, 𝜆 𝑖 is the principal stretch related to the principal nominal strain 𝜀 𝑖 (𝜆 𝑖 = 1 + 𝜀 𝑖 ), J is the volume ratio (J=𝜆 1 𝜆 2 𝜆 3 ), and 𝜇 𝑖 , 𝛼 𝑖 and 𝛽 𝑖 are the material parameters Specifically, 𝜇 𝑖 is the shear modulus, which should be positive and is related to the initial shear modulus 𝜇 0 according to

The initial bulk modulus 𝐾 0 is obtained from

In each term of the energy function, 𝛽 𝑖 determines the degree of compressibility 𝛽 𝑖 is related to Poisson’s ratio 𝑣 𝑖 by the equation

𝑣 𝑖 can generally be directly obtained in an experimental manner Thus, 𝛽 𝑖 can be calculated using equation (2.13)

The parameters 𝜇 𝑖 and 𝛼 𝑖 normally cannot be predicted by the properties of the foam, the density, or the deformation mechanism Instead, they must be identified by a curve-fitting process using experimental and combinations of the variables from the material test results Caliskan applied uniaxial compression, planar shear, simple shear, and volumetric compression test results to establish the PU material model for a jounce bumper analysis This model indicated considerable deviation between the calculated uniaxial compression curves and the experimental uniaxial compression curves However, the calculated values and the experimental values coincided if only the uniaxial compression data were used Because uniaxial compression is the primary deformation of the jounce bumper, only uniaxial compression data were applied in this study to fit the curve In uniaxial compression, the deformation mode is based on a single form of the nominal stress–stretch relation

𝑁 𝑖=1 (𝜆 𝐿 𝛼 𝑖 - 𝐽 −𝛼 𝑖 𝛽 𝑖 ) (2.14) where 𝑇 𝐿 is the nominal stress along the load direction and 𝜆 𝐿 is the stretch along the load direction In this equation, the nominal stress is obtained from the derivative of the strain energy function with respect to the principal stretch ratio The material coefficients are identified by a curve-fitting process if the stress–strain relationship is known from the uniaxial test The number of test data groups should be at least twice the order of the strain energy function

The material coefficients can be determined from the experimental data by a least- squares fitting process that minimizes the relative error in stress For K (≥ 2N) pairs of nominal stress–strain data, let 𝑇 𝑘 𝑡𝑒𝑠𝑡 be the nominal stress from the test data, and 𝑇 𝑘 𝑡ℎ be the nominal stress derived from foam model Hence, the relative error is

The least-squares criterion requires that ∑ 𝐸 𝑘 2 is minimized A vector x of order 2N is then introduced according to x = (𝑥 1 , 𝑥 2 , … , 𝑥 2𝑁 ) 𝑇 = (𝜇 1 , 𝛼 1 , … , 𝜇 𝑁 , 𝛼 𝑁 ) 𝑇 (2.16)

The strain energy function is nonlinear with respect to 𝛼 𝑖 Therefore, a nonlinear least-squares fitting process is required The Levenberg–Marquard algorithm was applied to fit the data This algorithm iteratively calculates a set of vectors 𝑥 (𝑟) starting from an initial vector 𝑥 (0) The vector 𝑥 (𝑟+1) is derived from the vector 𝑥 (𝑟) using the equation

𝑥 𝑖 (𝑟+1) = 𝑥 𝑖 (𝑟) - ∑ 6 𝑗=1 ∑ 𝐾 𝑘=1 (𝑃 𝑖𝑘 (𝑟) 𝑃 𝑗𝑘 (𝑟) + 𝛾𝛿 𝑖𝑗 ) −1 𝑃 𝑗𝑘 (𝑟) 𝐸 𝑘 (𝑟) (2.17) where r is the iteration count, 𝛾 is an algorithm parameter, and 𝑃 𝑖𝑘 is the derivative of the relative error 𝐸 𝑘 with respect to the coefficients 𝑥 𝑖 as given by

The coefficients are updated and obtained by repeating the iteration procedure defined by equation (2.17).

Theoretical basis of finite element method and Dynamic Explicit

2.2.1 Theoretical basis of finite element method

The finite element method (FEM) is the best performance numerical method for finding the approximate shape of the unknown function in domain V This method does not find an approximate of the unknown function on the whole domain, but only considers on each sub-domain V e Therefore, the FEM is very useful for many physical and engineering problem In these problems, the function is determined on the complicated domain which contains the sub-domain with the different geometry and boundary condition

In the FEM, the whole domain V will be divided into small pieces Those are called " Finite Element" Those elements connect all characteristic points (called Nodes) that lie on their circumference A node is simply a point in space, defined by its coordinates, at which degree of freedom (DOF) are defined Infinite element analysis, a degree of freedom can take many forms but depends on the type of analysis being performed For instance, in structural analysis, the degrees of freedom are displacements (Ux, Uy and Uz), while in a thermal analysis the degree of freedom is the temperature (T) These field variables are calculated at every node from the governing equation Field variable values between the nodes and within the elements are calculated using interpolation functions, which are sometimes called shape or base functions

To understand how to Finite Element Method works, the main steps of FEM solution are described below:

1 Discretize the continuum: the first step is to divide the region V into finite elements Ve For each problem, the type of element will be corresponding determined There are of course many types of element, covering the complete range of spatial dimension The most common are shown in table 2.2

Table 2.1 Simple geometry of the elements

First order element Second order element

2 Choose the interpolation functions: Interpolation functions are used for interpolating the field variables over the element Normally, the polynomials are selected

Then represent the approximate function according to the set of values and possibly its derivatives at the nodes of the element {qe}

3 Establish the element matrices which include stiffness element matrices [K]e, load element matrices {P}e, …

The result obtained can be formally represented as an element equation:

4 Assemble the element matrices to the global matrix: To achieve the global matrix system for the whole domain solution, the assembly of element matrices must be done After assembling, the boundary conditions (which are not accounted in element equation) will be imposed

5 Solve the global equation system (2.20): After imposing the boundary conditions, the global equation system will be computed by direct or iterative methods

With linear problems solving algebraic equations is not difficult The results are found the displacements of the nodes

But for the nonlinear problem, the solution will be obtained after a series of iterations that after each step the stiffness matrix changes (in the nonlinear physics problem) or the node force vector changes (in the geometric nonlinear problem) Calculate additional results: Sometimes, additional parameters need to be computed For instance, in mechanical problem, the stress or strain values are found according to the displacement values which are obtained after solving the global equation system Calculate additional results: Sometimes, additional parameters need to be computed For instance, in mechanical problem, the stress or strain values are found according to the displacement values which are obtained after solving the global equation system

2.2.2 Theoretical basis of Dynamic Explicit [28]

The basic equations solved by an Explicit Dynamic analysis express the conservation of mass, momentum and energy in Lagrange coordinates These, together with a material model and a set of initial and boundary conditions, define the complete solution of the problem

For Lagrange formulations, the mesh moves and distorts with the material it models, so conservation of mass is automatically satisfied The density at any time can be determined from the current volume of the zone and its initial mass:

The partial differential equations which express the conservation of momentum relate the acceleration to the stress tensor 𝜎 𝑖𝑗 : p𝑥̈ = 𝑏 𝑥 + Ə𝜎 𝑥𝑥 Ə𝑥 + Ə𝜎 𝑥𝑦 Ə𝑦 + Ə𝜎 𝑥𝑧 Ə𝑧 (2.22) p𝑦̈ = 𝑏 𝑦 + Ə𝜎 𝑦𝑥 Ə𝑥 + Ə𝜎 𝑦𝑦 Ə𝑦 + Ə𝜎 𝑦𝑧 Ə𝑧 (2.23) p𝑧̈ = 𝑏 𝑧 + Ə𝜎 𝑧𝑥 Ə𝑥 + Ə𝜎 𝑧𝑦 Ə𝑦 + Ə𝜎 𝑧𝑧 Ə𝑧 (2.24) Where:

Conservation of energy is expressed via:

𝑝 (𝜎 𝑥𝑥 𝜀 𝑥𝑥 ̇ +𝜎 𝑦𝑦 𝜀 𝑦𝑦 ̇ + 𝜎 𝑧𝑧 𝜀 𝑧𝑧 ̇ + 2𝜎 𝑥𝑦 𝜀 𝑥𝑦 ̇ + 2𝜎 𝑦𝑧 𝜀 𝑦𝑧 ̇ + 2𝜎 𝑧𝑥 𝜀 𝑧𝑥 ̇ ) (2.25) For each time step, these equations are solved explicitly for each element in the model, based on input values at the end of the previous time step

Only mass and momentum conservation is enforced However, in well posed explicit simulations, mass, momentum and energy should be conserved Energy conservation is constantly monitored for feedback on the quality of the solution (as opposed to convergent tolerances in implicit transient dynamics)

The Explicit Dynamics solver uses a central difference time integration scheme (Leapfrog method) After forces have been computed at the nodes (resulting from internal stress, contact, or boundary conditions), the nodal accelerations are derived by dividing force by mass:

𝑚 + 𝑏 𝑖 (2.26) where 𝑥 𝑖 ̈ are the components of nodal acceleration (i=1,2,3), 𝐹 𝑖 are the forces acting on the nodes, 𝑏 𝑖 are the components of body acceleration and m is the mass of the node

With the accelerations at time n ‐ ẵ determined, the velocities at time n + ẵ are found from:

𝑥̇ 𝑖 𝑛+1/2 = 𝑥̇ 𝑖 𝑛−1/2 + 𝑥̇ 𝑖 𝑛 Δ𝑡 𝑛 (2.27) Finally the positions are updated to time n+1 by integrating the velocities:

Advantages of using this method for time integration for nonlinear problems are:

• The equations become uncoupled and can be solved directly (explicitly) There is no requirement for iteration during time integration

• No convergence checks are needed since the equations are uncoupled

• No inversion of the stiffness matrix is required All nonlinearities (including contact) are included in the internal force vector

2.2.3 Theoretical basis of Dynamic Implicit [28]

The basic equation of motion solved by an implicit transient dynamic analysis is

𝑚𝑥 ̈ + 𝑐𝑥 ̇ + 𝑘𝑥 = 𝐹(𝑡) (2.29) Where m is the mass matrix, c is the damping matrix, k is the stiffness matrix and F(t) is the load vector

At any given time, t, this equation can be thought of as a set of "static" equilibrium equations that also take into account inertia forces and damping forces The Newmark or HHT method is used to solve these equations at discrete time points The time increment between successive time points is called the integration time step

• Implicit time integration is unconditionally stable for certain integration parameters

• The time step will vary only to satisfy accuracy requirements

• The solution is obtained using a series of linear approximations (Newton‐Raphson method), so each time step may have many equilibrium iterations

• The solution requires inversion of the nonlinear dynamic equivalent stiffness matrix

• Small, iterative time steps may be required to achieve convergence

• Convergence tools are provided, but convergence is not guaranteed for highly nonlinear problems

2.2.4 Comparison of Implicit and Explicit Methods

Implicit and Explicit analysis dissociate in the approach to time incrementation In Implicit analysis, each time increment has to converge, but users can set quite long time increments Conversely, explicit analysis doesn’t have to converge each increment, but time increments must be super small for the accuracy of solution

Figure 2.2 Comparison of Implicit and Explicit solution costs

The implicit solver is really useful if conditions in your analysis happen relatively slow Ideal analyses are generally longer than 1 second The advantage is that the time increment can be set as big as users want:

1 Unknown x is found by inversion of stiffness matrix (K)

2 Newton-Raphson / Enforced equilibrium solutions

3 Since global equilibrium is verified at each time increments, those increments can be large

4 Each time, increments are computed slowly and needed to get to the global equilibrium

6 Good decision for structure dynamic types which load rates are slow:

The explicit solver is great for fast events (mostly less than 0.1 second) Explicit solver calculates how big the time increment should be and set the time increment The speed of sound in material properties depends on density of material This is called “mass scaling” The time step in explicit depends on the mesh (element size and element quality), young modulus, and density

1 Unknown x is found by inversion of mass matrix (M)

3 Central difference methods are used

4 Each increment is calculated extremely fast

5 The time step has to be super small Otherwise, it’s impossible to pursue this equilibrium

7 Good decision for structure dynamic types which are wave propagation;

➢ From difference between Implicit and Explicit Dynamic so some problems (Drop Test) need to be surveyed in thesis are suitable with Explicit Dynamic.

Criterions for evaluating the energy absorption capacity of structure

In this review, the crashworthiness criteria used to evaluate the performance of corrugated structures are defined based on the force- displacement response of an energy absorber under crushing force A typical force-displacement response is presented in Figure 2.2 Although crashworthiness criteria such as initial peak crushing force, energy absorption, mean crushing force, crushing efficiency, specific energy absorption and undulation of load-carrying capacity, energy efficiency have been heavily reported in the literature, systematic summary of these criteria is rare Here, crashworthiness criteria are systematically summarized to provide better understanding of corrugated energy-absorbing structures as well as other energy absorbers In this review, we include indicators of

“deformation efficiency”, “structural effectiveness” and “energy-absorbing effectiveness”, which are effectively adopted to evaluate the performance of circumferentially corrugated tubes The key performance indicators are defined here:

Initial peak crushing force (IPCF): The peak force at an early stage in the crushing process of a structure

Initial peak crushing force (IPCF): The peak force at an early stage in the crushing process of a structure Energy absorption (EA): EA is mainly used to evaluate an energy absorber’s ability to dissipate crushing energy through plastic deformation It can be calculated as follows:

0 (2.30) where F(s) is the crushing force as a function of displacement s during the crushing process, 𝑙 𝑚𝑎𝑥 is the effective deformation distance (or effective stroke)

Figure 2.3 Force-displacement, energy-displacement and deformation efficiency- displacement curves of an axially loaded tube

Deformation Efficiency (f): The ratio of energy absorbed to the theoretical maximum force in the crushing distance:

(2.31) where s is the axial crushing distance, F(s) is the axial crushing force, 𝐹 𝑚𝑎𝑥 denotes the maximum crush force in the interval [0, s] Theoretically, the deformation efficiency increases with the crushing distance and achieves maximum value at the effective deformation distance On the basis of the drop in deformation efficiency, the 𝑙 𝑚𝑎𝑥 can be determined (Figure 2.2)

Crushing efficiency (𝑆 𝑒 ): Defined as the ratio of the effective stroke to the total length L of the absorber under compression, it characterises the effective utilisation rate of the material being used in this energy absorber:

Mean crushing force (𝑃 𝑚 ): The average compressive force exerted by the energy absorber over the total effective deformation It is defined as:

Crushing Force Efficiency (CFE): Defined as the ratio of 𝑃 𝑚 and IPCF It is a measure of load fluctuations that occur during the crushing of the structure:

Specific energy absorption (SEA) per unit mass: Defined as:

(2.35) where m is the mass of the energy absorber The SEA is often used to compare the energy-absorbing ability of different materials and structures

Specific energy absorption per unit volume (SEAv):

(2.36) where V is the volume of the energy absorber The SEAv is often used for sandwich structures

Undulation of load-carrying capacity (ULC): Defined as the ratio between the work done by the deviation of the actual crush force from the mean crush force and the total energy absorption within the effective stroke 𝑙 𝑚𝑎𝑥 This indicator is used to evaluate the uniformity of the force-displacement response of the energy absorber Energy absorbers with low ULC show better energy absorption efficiency:

Energy efficiency (𝐸 𝑒 ): Defined as the ratio of energy absorbed to the theoretical maximum energy that can be absorbed:

Structural effectiveness (η): Defined as the ratio of 𝑃 𝑚 and the maximum force that a structure can afford at the ultimate strain: η= 𝑃 𝑚

(2.39) where A is the cross-sectional area of a thin-walled section 𝜎 𝑢 is the ultimate strain of the material For a perfectly plastic material 𝜎 𝑢 = 𝜎 𝑦 (𝜎 𝑦 is the yield strain) η can be written as 𝑃 𝑚 /𝐴𝜎 𝑦

Energy-absorbing effectiveness factor (ψ): defined as the ratio: ψ = 𝑡𝑜𝑡𝑎𝑙 𝑒𝑙𝑎𝑠𝑡𝑖𝑐 𝑎𝑛𝑑 𝑝𝑙𝑎𝑠𝑡𝑖𝑐 𝑠𝑡𝑟𝑎𝑖𝑛 𝑒𝑛𝑒𝑟𝑔𝑦 𝑎𝑏𝑠𝑜𝑟𝑏𝑒𝑑 𝑏𝑦 𝑎 𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝑚𝑒𝑚𝑏𝑒𝑟

(2.40) For static loading, factor ψ can be written as: ψ = 3𝑃 𝑚

4𝜎 0 𝐴  𝑟 (2.41) where  𝑟 is the rupture strain The factor ψ is normally used for comparison of the efficiency of tubes made from different materials

The physical characteristics of various falls and associated calculations related to fall dynamics and force of impact (potential severity) illustrate the critical importance of fall prevention and the use of various fall protection control structures, devices, and activities.

• Velocity upon impact (v): v = (𝑣 0 2 + 2𝑔𝑠) 1/2 or v = √(𝑣 0 2 + 2𝑔𝑠) (2.42) where: v = velocity upon impact (ft/s)

𝑣 0 = initial velocity (ft/s) g = acceleration due to gravity (32.2 ft/𝑠 2 ) s = distance of the fall (ft)

2𝑑 (2.43) where: a = rate of deceleration (ft/𝑠 2 ) v = the velocity at the point of impact (ft/s) d = deceleration distance (ft)t

• G-Force (G) (Conversion of rate of deceleration to G-Force):

G = G-Force a = rate of deceleration (ft/𝑠 2 ) g = acceleration due to gravity (32.2 ft/𝑠 2 )

𝐹 𝑖 = force of impact (pounds force)

W = object weight (lbs) a = rate of deceleration (ft/𝑠 2 ) g = acceleration due to gravity (32.2 ft/𝑠 2 )

𝑃 𝑖 = pressure of impact (force per unit area in lbs/𝑖𝑛 2 )

𝐹 𝑖 = force of impact (pounds force)

• Ability of objects to withstand force of impact:

F= sA (2.47) where: s = stress absorption characteristics of the impacted material

A = the area over which the force is applied

Regarding the ability of objects to withstand a force and pressure of impact without injury, one must compare the induced stress of impact (force of impact) to the tensile, compression, shear, puncture, and bending stress that the object being impacted can withstand without damage Injury to the human body can occur due to direct impact forces, or due to transferred energy to underlying structures when such impact force is transferred to other elements of the body, such as muscles, ligaments, bones, and joints The time available for the absorption of impact forces (the absorption rate) will also affect the degree of injury.

ENGINERRING PROBLEMS

Energy absorption of Foam in drop test (Measurement Tool)

Figure 3.15 Fall drop model of Measurement Tool on Ground from height 1m

1 Observe deformation and Energy Absorption of Foam with SP500 Material

2 From material datasheet of customer provided, how to convert and calculate to have stress-strain curve for SP500 Foam material

3 From Actual Test about Acceleration graph of product, Simulation matched with Testing with the deviation is allowed below 10%

1 Drop a product (Measurement Tool) from height 1m with incline 2-2deg as figure

2 Using Low Density Foam Model for Foam Material as proved at problem 3.1

3 Boundary condition: Ground is fixed in 6 directions

4 Perform Acceleration graph matching between Simulation with Testing

3.3.1 Detail geometry of Hulk product

Figure 3.16 Cross Section view of Measurement Tool product

Figure 3.17 Other view of Measurement Tool product

Figure 3.18 Introduction about SP500 Material

Figure 3.19 Material properties of SP500

Figure 3.20 Energy Absorption of SP500 Material 3.3.3 Convert and calculate to have stress-strain curve for SP500 Foam material

Based on Brochure Material Properties and Impact Insulation EN of Sylodamp [33], we can calculate to have stress – strain curve as table below:

Table 3.1 Some formulas to calculate and have stress – strain curve

Calculation of impact force for an elastic Sylodamp bedding

Max deformation (mm) s =  x 12.5 (Material thickness)

From Table 3.1, we have stress-strain curve as below:

Figure 3.21 The stress-strain curve for SP500 material

Figure 3.22 Set up model for Measure Tool

1 The G-Force time graph and deformation of SP500 material:

At fully impact (1.6e-3), the maximum G-Force is 930G

Figure 3.23 The G-Force time graph and deformation of Foam SP500

Figure 3.24 Foam compression in drop process of Foam SP500

3 Behavior of components when having support from Foam:

Figure 3.25 Behavior of components when having support from Foam SP500

Product is dropped in the Z direction with height 1m in reality:

Figure 3.26 The G-Force time graph from Testing

(reference from testing data in company)

3.3.7 Compare G-Force time graph between Simulation with Physical drop

Figure 3.27 Compare G-Force time graph between Simulation with Physical drop

3.3.8 Compare G-curve matching between Simulation with Physical drop

Table 3.2 Compare G-curve matching between Simulation with Physical drop

There are 4 criterions to compare result between Simulation with Physical drop respectively:

+ Peak G-force value and duration: matched (allowed deviation < 10%)

+ Foam compression: simulation is approximately with Physical drop

+ Behavior: No impact happened in both Simulation and Physical drop

From 4 criterions as above, with current simulation result, stress/ strain on model can be used to evaluate.

Compare affection of some Foam materials: SP100, SP300, SP500 and SP1000

Figure 3.28 Fall drop model of Measurement Tool on Ground from height 1m

1 Observe deformation and Energy Absorption of Foam with some Foam materials

2 From material datasheet of customer provided, how to convert and calculate to have stress-strain curve for any Foam material

3 Make the suitable Foam material choice for product

1 Drop a product (Measurement Tool) from height 1m with incline 2-2deg as figure

2 Using Low Density Foam Model for Foam Material as proved at problem 3.1

3 Boundary condition: Ground is fixed in 6 directions

3.4.1 Foam material properties (datasheet is provided by customer):

Figure 3.31 Energy Absorption of SP100 Material

3.4.2 Convert and calculate to have stress-strain curve for some Foam materials

Similar to 4.3.3, based on Brochure Material Properties and Impact Insulation EN of Sylodamp [33], we can also calculate to have stress – strain curve for SP100, SP300, SP1000 as table below:

Table 3.3 Some formulas to calculate and have stress – strain curve

Calculation of impact force for an elastic Sylodamp bedding

Max deformation (mm) s =  x 12.5 (Material thickness)

At fully impact (1.4e-3), the maximum G-Force is 925G but SP100 is quite soft due to some elements are distorted so Simulation could not be completed

3.4.4 Compare G-Force – Time duration and Energy Absorption for some Foam materials:

Based on G-Force graph and Energy Absorption graph of some Foam materials, we can conclude that the lower the Energy Absorption, the more G-Force increases:

Figure 3.50 Compare G-Force Time and duration graph of some Foam materials

Figure 3.51 Compare Energy Absorption – Time graph of some Foam materials

3.4.5 Compare deformation of some Foam materials:

Figure 3.52 Compare deformation of some Foam materials

Based on deformation result of some Foam materials, SP300 is absorbed the most energy while SP1000 is the least

3.4.6 Simulation summary for some Foam materials:

Table 3.4 Suggest suitable Foam material for Measurement Tool

With simulation result of some Foam materials based on 4 criterions: Peak G-force value and duration, Foam compression, Behavior and Inclined angle, we can conclude:

+ With SP100, hard impact happened so SP100 need to be ignored

+ With SP1000, peak G-Force value is 1300G, larger than allowed G-Force value (

Tiêu đề	Assessment of energy absorption of Foam Material for applications in fall protection by Finite Element Analysis
Tác giả	Vo Thanh Binh
Người hướng dẫn	Assoc. Prof. Dr. Vu Cong Hoa, Dr. Nguyen Ngoc Minh
Trường học	Ho Chi Minh City University of Technology
Chuyên ngành	Engineering Mechanics
Thể loại	Master’s Thesis
Năm xuất bản	2024
Thành phố	Ho Chi Minh City

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