This is the second question of my homework: QUESTION 2 i. Evoke the expression r

Question

This is the second question of my homework: QUESTION 2 i. Evoke the expression relating true stress and true strain. What is the significance of the strain hardening exponent in product development? ii. Assume that for a tensile test, d?t/d?t = ?t when necking begins. Based on this condition, determine the true strain at the onset of necking? Explain carefully any assumptions made. iii. Evoke the expression for resolved shear stress. Next with your expression, explain critical resolved shear stress.

Explanation / Answer

i) True stress = (engineering stress) * exp(true strain) Strain Hardening exponent : In the formula s = K e n, s represents the applied stress on the material, e is the strain and K is the strength coefficient. The value of the strain hardening exponent lies between 0 and 1. A value of 0 means that a material is a perfectly plastic solid, while a value of 1 represents a 100% elastic solid. Most metals have an n value between 0.10 and 0.50. ii)General Concept: The strength of strain hardening materials continually increases with increasing deformation. At necking the load carrying capacity starts to decrease ( load carrying ability decreases, strength which is a stress continues to increase). The load carrying capacity decreases because the effect decreasing cross-section area overcomes the effect of increasing material strength due to strain hardening To identify the onset of necking the load, P, is used. When necking starts the load starts to decrease. Engineering stress is defined in terms of the specimen initial cross-section area, seng = P / Ao. So, a decrease in P results in a decrease in seng since Ao is constant. The onset of necking can be associated with the start of decreasing stress on the engineering stress-engineering strain plot. Since there is a relation between engineering strain and true strain, the true strain at necking can be found if the engineering strain at necking is known. Model of Instability in Tensile Deformation The model development is based on describing the load carrying behavior at the maximum load state. The load P is supported over cross section area A by stress s P = s A Since the question posed is to find the strain at necking, the area A in the force description is put in terms of tensile true strain e e = ln ( l / lo ) for constant volume, V volume at any time V = initial volume Vo l (Pi/4) D² = lo ( Pi/4) D²o Do² / D² = l / lo e = 2 ln ( Do / D ) ln ( Do / D ) = e / 2 D = Do exp[ -(e/2) ] A = (Pi/4) D2 = (Pi/4) D²o exp( -e ) P = s (Pi/4) D²o exp( -e ) The maximum load point has zero slope and the description of this is dP / de = 0 Using the expression for P above and checking the second derivative the maximum load condition is dP / de = (Pi/4) D²o { ds / de [ exp( -e ) ] - s exp( -e ) } = 0 ds / de = s This general result can be applied to any material. With the specific material behavior s = K en ds / de = s becomes n K e(n-1) = K en The result is e = n For the particular material behavior used localized deformation in tension starts when the strain is numerically equal to the strain hardening exponent. iii)Resolved shear stress is given by t = s cos F cos ? where s is the magnitude of the applied tensile stress, F is the angle between the normal of the slip plane and the direction of the applied force and ? is the angle between the slip plane direction and the direction of the applied force. whereas, critical resolved shear stress value is given by t =s (cosF cos?)max Resolved shear stress is the shear component of an applied tensile (or compressive) stress resolved along a slip plane that is other than perpendicular or parallel to the stress axis. The critical resolved shear stress is the value of resolved shear stress at which yielding begins; it is a property of the material.

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