3. Stress-Strain State Model

Consider the section A-A of the rod consisting of two regions (Figure 2(а)).

The outside Area I corresponds with cured zone. The inner Area II corresponds with zone where resin with filler is uncured.

Hooke’s law for orthotropic solid in cylindrical coordinates taking into account the chemical shrinkage is given by

(1)

(2)

(3)

where, and—radial, circumferential and axial stress components;

, and—radial, circumferential and axial strain components;

, and—modulus of elasticity of material in corresponding directions;

, , , , and—Poisson’s ratios;

, ,—linear thermal expansion coefficients in corresponding directions;

—temperature function;

, ,—chemical shrinkage of material in corresponding directions.

We considered the outside Area as transversely isotropic cylindrical solid. For this solid

(4)

Chemical shrinkage of the material along the fibers is neglected, i.e..

Also under symmetry of elastic constants the next relations are take place:

. (5)

For generalized plane strain state

(6)

The equilibrium equation for infinitely small element inside cylinder in local cylindrical coordinates (z direction—along fibers) for plane is follow:

(7)

Geometrical relations

(8)

It is accepted that inner Area II of the rod is acting like incompressible liquid. The liquid resin is pressurized and acts on the walls of cured cylinder (Figure 2(b)). Then boundary conditions for cylinder are following

for for (9)

Solve Equations (1)-(3) by stresses, using (4)-(8) and boundary conditions (9) the following equations for stresses can be obtained:

(10)

(11)

(12)

The value of pulling force N we can find from (13):

(13)

where is the total resisting force per unit area of the composit in contact with the die and S the internal surface area of the die. The solution for (13) is described in [7].

4. Failure Criteria

As mentioned above, at high pull speeds the strain occurred in a polymerized part of the rod may exceed the maximum allowable values for the material which leads to the main crack appearance. As a strength criterion the maximum deformation criterion is chosen which has the form [8]:

(14)

The study determined that the values of the stresses in the cross section of the rod are far from the limit values, whereas the strain exceeds the limiting values at high pull speed. Therefore, the maximum deformation criterion was chosen as the primary. The sign “+” means tension, “−” compression, the ultimate strain obtain as

(15)

there—experimentally determined limiting values of stresses for a material by transverse tension, transverse compression and tension along the fibers, respectively. The strains are defined by (1)-(3) with subject to (10)-(12) and compared with ultimate values from (15).

5. The Heat Transfer and Polymerization Model

For stress-strain state determination we need to know the temperature field distribution and location of the border between cured and liquid material. For this we should solve the non-linear heat transfer and polymerization problem which for pultrusion of the rod has a view [9]:

(16)

(17)

(18)

where C—heat capacity; ρ—density; V—pulling velocity; W—reaction heat; λ—thermal conductivity; ψ—degree of polymerization; χ—resin volume; T_t—absolute temperature; T₀ —rod temperature at inlet of die; K₀—preexponential factor; E_A —active energy of process activation; R—gas constant; n—total reaction order by reacting components; γ—heat-transfer coefficient;—ambient temperature.

Based on the solution of (16)-(18) by the method of finite differences [11] the radius R₁ of the inner zone II is obtained. For this radius the degree of polymerization ψ = 0.95 (Figure 3(a)).

Using the value R₁ the interval of the temperature distribution over the cross section is determined for the calculation of integrals in the (10)-(12) (Figure 3(b)). The stress values are numerically obtained using the trapezium method.

6. Pressure Model

One of the process parameter in pultrusion is the pressure-rise inside a die. The pressure of liquid resin appears due to fiber/resin system passes through the die (Figure 4). The equation of incompressibility [5,6] for the flow of resin moving through the fibers in a cylindrical coordinate system is follow:

(19)

In the projections of the velocity vectors onto the axis z (along the rod axis)

(20)

 In the projections of the velocity vectors onto the axis r (in radial direction)

(21)

here K₁₁—medium permeability in axial direction; K₂₂— medium permeability in radial direction; v—component of resin velocity in radial direction; u—component of resin velocity in longitudinal direction; V—component of fiber velocity in radial direction; U—component of fiber velocity in longitudinal direction (pull speed); μ—viscosity;—porosity; V_f —the function of changes in the fiber volume fraction along the length of the tapered die.

Expressions (19)-(21) are based on Darcy’s law for the resin flow, passing through a porous medium (fiber) at low speeds. According to Gebart’s model [10] the permeability in the axial and transverse directions are given as

(22)

where R_f is the fiber radius, c = 53, C₁ = 0.231 and V_fmax = 0.9068.

Taking into account (20) and (21) and that

where—geometrical parameter of the input section of the die (the taper angle), is the local radius of the die wall, we can obtain the following differential equation for the pressure from (19):

(23)

where

V_s—is the fiber volume fraction in the straight portion of the die (or finished products), R_p is the aperture radius of the last pre-form plate.

To solve the problem (23) by finite differences method [11], the solution region divided into four zones (Figure 4). In this only half die simulated due to symmetry. It was assumed that the zone I completely filled a resin (it is a resin backflow region there excess resin is squeezed out of the die due to the compression of the fiber/resin system). For this region Equation (23) has a view:

(24)

For a region II therefore (23) can be written as

(25)

For a III region the Equation (23) is applied in general form. For a IV region the Equation (25) is used taking into account that

At the computational domain inlet, the pressure is considered to be equal to atmospheric pressure P₀ (101,325 Pa). The boundary conditions on the tapered wall of the die were determined from the ratio of velocity components at the boundary:. The boundary condition for the grid points of the rod, where the polymerization is completed ():. At the center line (r = 0), and At the Top boundary (straight portion of die):.

7. Results and Discussion

Thus, solving the system of Equations (10)-(18), (23)-(25) we can obtain the strains for every rod cross section at the outlet of the die and, comparing them with ultimate values, prevent main crack appearance.

The following parameters were used when solving the problem. The rod radius R_s = 38 mm, pull speed 70 mm/min. Pre-form plate radius R_p = 1.085R_s. Fiber radius R_f = 0.013 mm, fiber volume fraction V_s = 0.6, taper angle α = 25⁰, length of the taper part of the die L1 + L2 = 76 mm, the resin viscosity under normal conditions Pa^*s, P₀ = 101325 Pa. The Poisson’s ratios; The thermal expansion coefficients 1/С⁰ , 1/С⁰ , Modulus of elasticity MPa, MPa. The next strength data for composit material are used [8]: longitudinal tensile strength MPa, transverse tensile strength MPa, transverse compressive strength MPa. Die temperature settings (the length of straight portion of the Die is 1 m): 110-150-190-160 С⁰. The parameters such as preexponential factor, reaction heat, active energy of process activation and total reaction order by reacting components was taken from [9].

As a result of solving the problem (23)-(25) the centerline pressure profile along the Die is obtained (Figure 5).

Figures 6 and 7 show the profiles of the radial (Figure

6) and circumferential (Figure 7) strains in rod section А-А (Figure (1)). It was assumed that external pressure (Figure 2 (b)), pulling force N. In this example the chemical shrinkage was taken as, the ultimate strain tension of composite material in according to (15) is.

To determine the maximum possible pull speed series of solutions at different pull speeds are obtained. For the each of the solution the maximum strains (circumferential and radial) were determined. These values are summarized in Table 1" target="_self"> Table 1. Figure 8 shows the maximum strain profiles as functions of pull speed.

8. Conclusions

Analyzing the results, it can be concluded that for this example the optimum pull speed is up to 1.33 mm/sec.

The numerical model is a useful tool to determine the stress-strain state of polymer composite materials at the outlet of the Die through the selection of appropriate values for the different process control parameters. An appreciable stress-strain state can help to suppress crack formation, and aid in the manufacture of a quality pultruded product.

REFERENCES