MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING CAMBRIDGE, MASSACHUSETTS 02139 2.002 MECHANICS AND MATERIALS II QUIZ I SOLUTIONS Distributed: Wednesday, March 17, 2004 This quiz consists of four (4) questions. A brief summary of each question’s content and associated points is given below: 1. (10 points) This is the credit for your (up to) two (2) pages of self-prepared notes. Please be sure to put your name on each sheet, and hand it in with the test booklet. You are already done with this one! 2. (20 points) A lab-based question. 3. (40 points) A multi-part question about a linear elastic boundary value problem. 4. (30 points) A ‘design for yield’ question. Note: you are encouraged to write out • your understanding of the problem, and • your understanding of “what to do in order to solve the problem”, even if you find yourself having “algebraic/numerical difficulties” in actually doing so: in short, • let me see what you are thinking, instead of just what you happen to write down.... The last page of the quiz contains “useful” information. Please refer to this page for equa- tions, etc., as needed. If you have any questions about the quiz, please ask for clarification. Good luck!Problem 1 (10 points) Attach your self-prepared 2-sheet (4-page) notes/outline to the quiz booklet. Be sure your name is on each sheet. Problem 2 (20 points) (Lab-Based Problem) In your own words, describ e the following terms used in the description of linear elastic stress concentration, and briefly illustrate (with schematic figures, simple equations, etc.) how these features are used, measured, or are otherwise identified: • (5 p oints) stress concentration factor A stress concentration factor, Kt, can be defined as the ratio of the [peak] value of a stress component,σlocal, at a highly-stressed location (e.g., a notch root) to a nominal, or far-field value of a stress component, σnom , that is [typically] readily associated with overall loading: σlocal Kt ≡ σnom ≥ 1. • (5 points) St. Venant’s principle St. Venant’s principle states that the perturbation in a nominally homogeneous stress state introduced by a [geometric/material] hetero-geneity (e.g., a hole, notch, cut-out, or reinforcement) of characteristic linear dimension “�” decays rapidly with distance from the heterogeneity. In practice, the effects of the perturbation in the stress field are negligible for distances greater than ∼ 3�. (10 points) Describ e “best practice” in the location of a resistance strain gauge to measure stress concentration at the root of a through-thickness notch in a planar body subjected to in-plane loading. (1 page, maximum). The key ideas here are to note that (a) a strain gauge records an average value of strain parallel to its direction, with the region sampled being that beneath the gauge, and (b), in the presence of strong stress concentration, there are steep gradients in the values of in-plane strain (and stress) in the immediate vicinity of the notch root. Thus, any attempt to use a laterally-mounted strain gauge in order to estimate the strain concentration “at” the root of the notch will be hindered by the inherent averaging-in of lower [than peak] strain in the region under the gauge. Conversely, viewing the body as having a small but finite thickness, t, at its notch root, the variation in strain along the notch is generally much less rapid than its variation “radially” away from the notch surface. Thus, a strain gauge mounted on the t-thick surface of the notch root provides a much more reliable estimate of the peak strain at the notch than can be obtained from a similar gauge mounted laterally on the faces of the body. 2Problem 3 (40 points) A large isotropic linear elastic body contains a long, embedded fiber, of radius a, extending parallel to the x3 axis (see Fig. 1, b elow). The fiber is made of a different material than the surrounding matrix; elastic moduli of the fiber greatly exceed those of the surrounding matrix. Thus, as an approximation, we can treat the fiber as a rigid, non-deforming body. Under a certain loading of the fiber-containing matrix, the spatial variation of the compo-nents ui of the displacement vector u in the matrix region is described by u1(x1, x2, x3) = 0 u2(x1, x2, x3) = 0 2�a� u3(x1, x2, x3) = γ x2 1 − 2x1 + x2 , 2 for a dimensionless constant “γ” satisfying γ � 1. (N.B.: These expression apply only in | | 2 2the matrix region exterior to the rigid fiber; that is, at locations satisfying x1 +x2 ≡ r2 .≥ a• (5 points) Evaluate the spatial dependence of all components of the strain tensor, �ij �11 ≡ ∂u1 ∂x1 = 0 �22 ≡ ∂u2 ∂x2 = 0 �33 �12 = �21 ≡ ≡ ∂u3 ∂x3 = 0 1 2 � ∂u1 ∂x2 + ∂u2 ∂x1 � = 0 �13 = �31 �23 = �32 ≡ ≡ 1 2 � ∂u1 ∂x3 + ∂u3 ∂x1 � = x1x2γa2 (x2 1 + x2 2)2 1 2 � ∂u2 ∂x3 + ∂u3 ∂x2 � = γ 2 �1 − a2 x2 1 + x2 2 + 2a2x2 2 (x2 1 + x2 2)2 � • (5 points)Based on the strain components calculated above, evaluate the spatial dependence of all components of the stress tensor, σij . 3E � ν � �ij +σij = (1 + ν) (1 − 2ν) �� �mm � δij . m=1 σ11 = σ22 = σ33 = σ12 = σ21 = 0; 3 2E 2x1x2Gγa2 σ13 = σ31 = �13 = 2G�13 = 2 21 + ν (x1 + x2)2 2 2E �a 2a2x2 �σ23 = σ32 = �23 = 2G�23 = Gγ 1 − 2 + 2 21 + ν x1 + x2 (x1 + x2)2 2 (5 p oints) Describ e the state of stress “far” from the fiber. Distance from the 2 2fiber can be measured by radius r ≡ �x1 + x2; when r/a � 1 (or, equivalently, when a/r � 1, the point is far from the outer boundary of the fiber. By introducing the