Fundamental Course in Mechanical Processing of Materials. Exercises

Transcription

1 Fundamental Course in Mechanical Processing of Materials Exercises 2017

2 3.2 Consider a material point subject to a plane stress state represented by the following stress tensor, Determine the principal stresses and directions using the Mohr circle. 5.1 Determine the engineering and true ultimate tensile strength of a material with the following true stress-strain curve:. 5.3 Consider a cold rolling operation of an annealed aluminium sheet (99% Al) that reduces the thickness by 15%. The true stress strain curve of the material is defined by σ = 140 ε 0.25 MPa. Determine the new yield stress of the material after being rolled. 5.5 Consider the cold cylindrical extrusion (direct) operation of a carbon steel rod with a stress-strain curve, σ = ε MPa. Calculate the increase in temperature resulting from the extrusion operation knowing that the final effective strain is ε = 1.2, and the density of the steel is ρ = 7.87x10 3 kg/m 3 and the specific heat c = 0.46x10 3 J/kg C. 1.2 The wire used in the manufacturing of a paper clip with 1 mm diameter is produced by wire drawing from a wire with an initial diameter of 10 mm. Calculate engineering strains and the true strains that the wire material undergoes in the longitudinal, radial and tangential direction during the wire drawing. 1.4 A round specimen with an initial gauge length of 50 mm and 13 mm of diameter is subjected to a uniaxial tensile force of 50 kn. Considering the plastic deformation is uniform and that the gauge length under load is 64 mm, determine: a) Engineering stress and strain b) True stress and strain c) The diameter of the specimen under load

3 1.5 A round specimen of a magnesium alloy with an initial gauge length of 30 mm and 12 mm of diameter is subjected to a uniaxial tensile test. The force and displacement values of the test are presented in the following table: Force F (kn) Displacement (mm) (max. force) (fracture) 2.61 The fracture gauge length is mm and diameter mm. a) Proceed to the graphical representation of the evolution of the force with displacement and the evolution of the engineering stress with engineering strain. b) Yield stress to 0.2% off-set. c) Engineering ultimate tensile strength. d) Young modulus for the magnesium alloy. e) Elongation. f) Area reduction. g) Fracture engineering stress. h) Fracture true stress. i) Resilience. 5.3 Consider a cold rolling operation of an annealed aluminium sheet (99% Al) that reduces the thickness by 15%. The true stress strain curve of the material is defined by σ = 140 ε 0.25 MPa. Determine the new yield stress of the material after being rolled. 5.5 Consider the cold cylindrical extrusion (direct) operation of a carbon steel rod with a stress-strain curve, σ = ε MPa. Calculate the increase in temperature resulting from the extrusion operation knowing that the final effective strain is ε = 1.2, and the density of the steel is ρ = 7.87x10 3 kg/m 3 and the specific heat c = 0.46x10 3 J/kg C.

4 1.2 The wire used in the manufacturing of a paper clip with 1 mm diameter is produced by wire drawing from a wire with an initial diameter of 10 mm. Calculate engineering strains and the true strains that the wire material undergoes in the longitudinal, radial and tangential direction during the wire drawing. 2.9 The sheet technological processes of plastic deformation are usually studied under conditions of plane stress. a) Identify the characteristic work area of these technological processes in the principal stresses space assuming that at least one of the principal stresses is of traction. b) Identify the highest tensile stress that can be supported by a sheet subjected to a biaxial tensile loading and determine the corresponding ratio. 4.1 Determine the minimum wall thickness of a tube closed at the ends with an average radius of 200 mm and 800 mm of length, so as to withstand an internal pressure of 10 MPa. For the design admits that this is a thin-walled tube and that the tube material has a yield strength of 300 MPa. Using the principal stresses plane discuss the results. Assume valid the von Mises yield criterion and analyse qualitatively the strains developed in the tube. 4.4 Determine the value of the effective stress and strain during a plane strain compression test, represented in the figure. b F w x y z h

5 2.3 The application of a triaxial stress state 80 MPa, 45 MPa and 30 MPa MPa on a metallic component produces an elastic deformation. Check that the component remain in elastic state when an additional hydrostatic loading with m 130 Mpa is applied Calculate the effective stress, effective strain increment and the increment of plastic work per unit volume for the following loadings: a) uniaxial traction. b) Plane strain deformation, with d 2 0 and An AISI 1015 steel cylinder with 20 mm diameter and 30 mm in height which is at room temperature ( T K ) is compressed between plates of greater extent than the part under adiabatic conditions and without friction. Use the ideal work method, assuming that the percentage of plastic work that is converted into heat 0. 85, to obtain the evolution of the cylinder temperature that is shown in the figure T (K) Additional informations of the AISI 1015 steel, Stress-strain curve, Density, 7870 kg/m 3 ; Specific heat, c 479 J/kg.K; p MPa;

6 3.2 A cylinder of C26000 brass (70 Cu-30 Zn) with 20 mm of diameter F and 30 mm in height is compressed between plates of greater extent than the workpiece and frictionless, in a hydraulic press with a constant velocity of v = 40 mm/s, until a final height of 8 mm is obtained. a) Calculate the compressive force at the final instant of the cold forming operation. h =30 0 b) Obtain the evolution of the compression force with the displacement of the upper plate. c) Determine the amount of energy needed to provide by the press to perform the entire compression operation. d) Provide an estimative of the amount of power of the hydraulic press required at the final instant of the compression operation. D =20 0 (mm) e) Calculate the compressive force at the final instant of the forming operation. Assume that the compression operation takes place at high temperature (600 C). Additional information regarding brass C26000: Stress-strain curve at low temperature (25 C), 500 ; Stress-strain curve at high temperature (600 C), 100 MPa; 3.5 An Aluminium AA1100-O rod with 150 mm of initial diameter is subjected to a cold direct extrusion operation in order to obtain a final diameter of 50 mm. The energy consumed in the redundant deformation is 40% of the energy required for the plastic deformation (ideal work), while the energy consumed by friction is 25% of the total energy required to perform the extrusion operation. a) Calculate the yield stress of the extruded rod. b) Calculate the extrusion force. Additional information regarding Aluminium AA1100-O: Mechanical behavior, 180 MPa.

7 3.6 A wire of 70CU, 30Zn annealed brass with 10 mm of diameter is subjected to a wire drawing operation in order to obtain a final diameter of 8 mm. grips draw plate d=8 mm T d 0 v=0.5 m/s The wire drawing speed is constant and of 0.5 m/s. The sum of the energy consumed to overcome friction and redundant deformation is equal to 40% of the energy required for plastic deformation (ideal work). a) Calculate the force of wire drawing,. b) Calculate the power required to perform the wire drawing. Additional information of the Brass 70CU-30Zn: Mechanical behaviour, 895 MPa. 4.1 Consider the compression operation with maximum friction ( k ) of a rectangular bar between plates with greater extent than the bar under plane strain conditions. y x z a) Calculate the value of the compression pressure dimensionless with the yield stress of the material p 2 k. b) Determine the maximum value of p 2 k. c) Determine the average value of p 2 k. d) Plot the variation of p 2 k with x distance. e) Plot the variation of the stresses x, y e z with x distance.

8 4.3 A cylindrical preform with 100 mm diameter and 25 mm in height is cold compressed under two friction conditions, 0 (frictionless) and R=50 h=25 (mm) a) Write the equation that permits calculate the value of the compression pressure, p at the beginning of the material plastic deformation for the two friction conditions. b) Compare the obtained equations with the equations that result from applying the ideal work method. c) Calculate the compressive force that must be applied to initiate plastic deformation on the material for the two friction conditions. Assume that the stress-strain behavior of the material of the preform can be approximated by a rigid perfectly plastic model where y 300 MPa. 4.4 Consider the compression operation with friction ( 0.3) between plates with greater length than the bar under plane strain conditions. The bar has 100 mm in length and a rectangular section of 20 mm width and 25 mm in height (thickness). a) Determine, without using the equation of the average pressure, the compressive force corresponding to 20% reduction in height. b) Determine, using the equation of the average pressure, the compressive force corresponding to 20% reduction in height. c) Sketch the variation in stress x, y e z inside the bar corresponding to 20% reduction in height, indicating the maximum and minimum values for each of the stresses. Additional information relating to the material: 0 5 Stress-strain equation of the material, 400. MPa.

9 14.1 Consider the cold forging operation of a non-symmetric crosshead that is shown in the figure (see 10.5). The preform used in the manufacturing of the crosshead is a cylinder of technically pure aluminium (99.95%) with a 14:54 mm of diameter and mm in height. The stress-strain curve of the material, was determined by uniaxial compression tests, is as follows: (MPa) Under these conditions, use the ideal work method to determine: a) The forging force considering that the projected area of the crosshead can be approximated by a square with sides of 29.5 mm. b) The energy required to forge the crosshead Consider the cold forging operation by axial compression of an Aluminium alloy 5000 series rod represented in the figure. The forging operation is lubricated having a friction factor of through ring tests. In these conditions: m obtained a) Establish a weighting pressure coefficient that is suited for the axial compression operations and determine the forging force corresponding to 60% of reduction of the free height of the rod. b) Calculate the required energy to forge the part for a).

10 7.1 Consider the hot closed die forging operation (1000º C) of a flange which is illustrated in the figure. The preform used in the manufacturing of the flange consists of a cylinder of AISI 1045 steel with 25 mm of diameter and 35 mm of height. a) Determine the required force to forge the flange in a hydraulic press with a constant velocity of v 10 mm/s considering that the projected area can be approximated to circle with 50 mm of diameter. Consider that Q p 3. b) Determine the power of the press in the final instant of deformation. c) Determine the time necessary to forge a part. Mechanical behaviour of the AISI 1045 steel at 1000 C: MPa. 7.4 Consider the compression operation between dies in plane strain deformation conditions of a bar with a square section of 50 mm side and 300 mm long shown in the figure. L =50 0 w=300 F h =50 0 y x z y 0 x h =50 0 (mm) a) Determine the maximum contact pressure of the dies, p max at the beginning of plastic deformation of the bar for two friction conditions, null friction and maximum friction ( k ). b) Calculate the compression force needed to start the plastic deformation of the bar, for the two friction conditions of question a). c) Calculate the compressive force that must be applied to reach a 50% reduction in height of the bar, considering maximum friction ( k ). Consider that the mechanical behaviour of the material can be approximated to a rigid-perfectly plastic behaviour with: 100 MPa. y

11 17.1 It is intended to produce by sheet metal cutting part that is shown in the figure. The manufacturing is performed in a progressive cutting tool from a coil of AISI 302 stainless steel with 3 mm of thickness. The ultimate tensile strength of the material is UTS 600 MPa. Determine the force and work necessary for the manufacturing of the part During the experimental stamping tests is usual to etch a circle grid in the surface of the blank in order to evaluate the deformation mode of the material during the stamping operation. This technique has the advantage of allow to determine the thickness variations in the critical areas of the parts. Consider a stamping operation where 3 mm initial diameter of circles was etched in a stamping steel sheet with 2 mm of thickness. In a particular area of the part experimental measurements were performed and the initial circle turned into an ellipse with the major axis and minor axis measuring respectively 3.4 mm and 3.2 mm.

12 The mechanical behaviour of the material is the following: 420 MPa. In these conditions and neglecting any anisotropy phenomena, determine for this point: a) The thickness of the blank after the deformation. b) The effective stress and strain values. c) The coefficient between the principal stresses 2 1. Consider a plane stress state and that the loading is proportional Cte Consider the operation of incremental forming of a sheet metal of aluminium alloy AA1050-H111, with 1.5 mm thick, which is shown in figure. The mechanical behaviour of the sheet material can be approximated by the following stress-strain 0.04 equation 140 MPa. Prior to the plastic deformation the sheet was etched with a grid of circles with 2 mm in diameter. At the end of the sheet plastic deformation it has been found the occurrence of a crack that is indicated in the figure having, in this region, the circles become ellipses with major and minor axes respectively equal 6.1 mm and 2.0 mm. Accordingly, the number for the region where the crack appears, a) Determine the principal strains on the surface of the sheet and represent their strain path in the plane of the principal strains 1, 2. b) Determine the value of the final thickness of the sheet. c) Knowing that the limit of incremental sheet forming operations is determined by the amount of reduction in the maximum thickness, quantify this limit and represent it in the principal strains plane 1, 2. d) Determine the effective stress and strain. Utilize the von Mises yield criterion. e) Determine the value of the coefficient of the principal stresses 2 1, considering a plane stress state 3 0. p f) Represent graphically the vector of the strain increment d ij in the plane of the principal stresses and characterize the type of deformation in this area of the part. g) Determine the value of the stresses 1 e 2.

13 Complementary Exercises

14 Chapter 14 Forging Problem 1 Consider the last forging operation of an AISI 1020 steel connecting rod that is shown in the figure (step 4 - finishing). The connecting rod is hot forged in in a mechanical press at a temperature of 1200 º C. It is estimated that the total volume of material required to fabricate the connection rod is equal to mm 3 (including an excess of 20% corresponding to the material that is ejected into the gutter in order to form the flash) The total projected area of the connecting rod in the last forging operation is equal to 5800 mm 2, and 2300 mm 2 corresponds to the projected area of the land and flash. The velocity of the upper die is equal to 0.25 m/s. a) Calculate the forging force using the ideal work method. Answer: F 4511 kn. b) Determine the required power at the final instant of the operation. Answer: P 1128 kw. Additional data: Mechanical behaviour of AISI 1020 steel (1200ºC): MPa

15 Chapter 17 Sheet Metal Cutting Problem 1 The following sheet metal part is to be fabricated by conventional blanking. The part must be manufactured in brass, supplied on sheets with 1 mm thick. a) Calculate the maximum cutting force assuming that the outer contour and the inner hole of the part are cut in a progressive tool with punches that have the same height. b) Perform a schematic representation of the evolution of the cutting force with the displacement of the punches. c) Establish the nominal dimensions of the punch and die that are used to cut the central hole of the part assuming a radial clearance equal to 7% of the sheet thickness. Additional data: Mechanical behaviour of Brass: R 380 MPa

16 Problem 2 A carbon steel sheet with 3 mm thickness and 2 m length is cut along a straight line. Calculate the shearing force under the following operating conditions: a) Using straight blades. b) Using blades with 6º inclination. Additional data: Mechanical behaviour of the carbon steel: R 450 MPa. Problem 3 A circular hole with 75 mm diameter is to be cut by blanking from an aluminium blank with 2 mm thickness. a) Setup the geometry of the punch and of the die considering a radial clearance equal to 10% of the blank thickness. Answer: d 75 mm and d mm. p m b) Supply the geometry of the punch and die if a circular part was to be fabricated instead of a circular hole. Answer: d mm and d 75 mm. p m c) Determine the maximum cutting force (consider C ). Answer: F kn. max d) Determine the cutting energy. Answer: W 130 J. e) Calculate the required power of the electric engine of the press if the cutting speed is approximately constant and equal to 150 mm/s and the efficiency is 70%. Answer: P kw. Additional data: Mechanical behaviour of the aluminium alloy: R 276 MPa

17 Chapter 22 Deep Drawing Problem 1 A cylindrical carbon steel cup with 150 mm diameter, 70 mm height and 1.2 mm thickness is fabricated by conventional deep drawing. a) Determine the geometry and dimensions of the blank. Answer: Circular blank with 254 mm diameter and 1.2 mm thickness. b) Calculate the maximum force assuming that the cup is produced in a single deep drawing stage. Answer: Fmax kn c) Having completed the fabrication of this cup, it was tried without success to produce a similar cup from a blank with 290 mm diameter and 1.2 mm thickness in a single deep drawing stage. Explain the reason why the operation was not successful. Answer: Fracture appears because m Suggestion: The values of the weighting coefficients from the tables 22.XI and 22.I of the book. Q f and limit drawing ratio must be obtained Additional data: Mechanical behaviour of the carbon steel: R 380 MPa