The elastic behavior of material for linear stress and linear strain, is shown in the figure. The energy density for a linear strain of 5 × 10^-⁴ is __________KJ/m³. Assume that material is elastic upto the linear strain of 5 × 10^-⁴.

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Raj Pandey | Contributor-Level 9

2 weeks ago

$Y = \frac{σ}{\in} = \frac{20}{1 0^{- 10}},$ Y → young’s modules of elasticity.

Stress $(σ)$ at strain $(\in)$ 5 × 10^-4

$σ = \in Y$

$= 5 \times 1 0^{- 4} \times 20 \times 1 0^{10}$

= $1 0^{2} \times 1 0^{6}$

= 10⁸

$U_{d} (E n e r g y d e n s i t y) = \frac{1}{2} σ \in$

$= \frac{1}{2} \times 1 0^{8} \times 5 \times 1 0^{- 4}$

$\begin{array}{l} = 2.5 \times 1 0^{4} \\ = 25 \times 1 0^{3} = 25 k J / m^{3} \end{array}$

<math> <mrow> <mi>Y</mi> <mo>=</mo> <mfrac> <mrow> <mi>σ</mi> </mrow> <mrow> <mo>∈</mo> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mn>0</mn> </mrow> <mrow> <mn>1</mn> <msup> <mrow> <mn>0</mn> </mrow> <mrow> <mo>−</mo> <mn>1</mn> <mn>0</mn> </mrow> </msup> </mrow> </mfrac> <mo>,</mo> </mrow> </math> Y → young’s modules of elasticity.              Stress <math> <mrow> <mrow> <mo>(</mo> <mrow> <mi>σ</mi> </mrow> <mo>)</mo> </mrow> </mrow> </math> at strain <math> <mrow> <mrow> <mo>(</mo> <mrow> <mo>∈</mo> </mrow> <mo>)</mo> </mrow> </mrow> </math>  5 × 10-4              <math> <mrow> <mi>σ</mi> <mo>=</mo> <mo>∈</mo> <mi>Y</mi> </mrow> </math>                <math> <mrow> <mo>=</mo> <mn>5</mn> <mo>×</mo> <mn>1</mn> <msup> <mrow> <mn>0</mn> </mrow> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> <mo>×</mo> <mn>2</mn> <mn>0</mn> <mo>×</mo> <mn>1</mn> <msup> <mrow> <mn>0</mn> </mrow> <mrow> <mn>1</mn> <mn>0</mn> </mrow> </msup> </mrow> </math>                = <math> <mrow> <mn>1</mn> <msup> <mrow> <mn>0</mn> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>×</mo> <mn>1</mn> <msup> <mrow> <mn>0</mn> </mrow> <mrow> <mn>6</mn> </mrow> </msup> </mrow> </math>                = 108              <math> <mrow> <msub> <mrow> <mi>U</mi> </mrow> <mrow> <mi>d</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>E</mi> <mi>n</mi> <mi>e</mi> <mi>r</mi> <mi>g</mi> <mi>y</mi> <mtext> </mtext> <mtext> </mtext> <mi>d</mi> <mi>e</mi> <mi>n</mi> <mi>s</mi> <mi>i</mi> <mi>t</mi> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mi>σ</mi> <mo>∈</mo> </mrow> </math>                <math> <mrow> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mo>×</mo> <mn>1</mn> <msup> <mrow> <mn>0</mn> </mrow> <mrow> <mn>8</mn> </mrow> </msup> <mo>×</mo> <mn>5</mn> <mo>×</mo> <mn>1</mn> <msup> <mrow> <mn>0</mn> </mrow> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> </math>                <math> <mtable columnalign="left"> <mtr> <mtd> <mrow> <mo>=</mo> <mn>2</mn> <mo>.</mo> <mn>5</mn> <mo>×</mo> <mn>1</mn> <msup> <mrow> <mn>0</mn> </mrow> <mrow> <mn>4</mn> </mrow> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mn>2</mn> <mn>5</mn> <mo>×</mo> <mn>1</mn> <msup> <mrow> <mn>0</mn> </mrow> <mrow> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> <mn>5</mn> <mi>k</mi> <mi>J</mi> <mo>/</mo> <msup> <mrow> <mi>m</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msup> </mrow> </mtd> </mtr> </mtable> </math>

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The elastic behavior of material for linear stress and linear strain, is shown in the figure. The energy density for a linear strain of 5 × 10^-⁴ is __________KJ/m³. Assume that material is elastic upto the linear strain of 5 × 10^-⁴.

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The elastic behavior of material for linear stress and linear strain, is shown in the figure. The energy density for a linear strain of 5 × 10-4 is __________KJ/m3. Assume that material is elastic upto the linear strain of 5 × 10-4.

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The elastic behavior of material for linear stress and linear strain, is shown in the figure. The energy density for a linear strain of 5 × 10^-⁴ is __________KJ/m³. Assume that material is elastic upto the linear strain of 5 × 10^-⁴.